Nurlybayev, S. Abdykarimova

ALGEBRA OF N-NOMIALS AND COMPLICATED PROBLEMS

(KazNPU after Abai, Almaty City)

Мақалада n-мүшелі шартты белгілерді (n-номдарды) дәрежеге ӛсіруі және биномдарды қысқартылған формулалар арқылы кӛбейтулері барынша талқылауы қарастырылады. n-номдардың бастапқы дәрежелері үшін формулаларының ықшамды тура қорытындылары кӛрсетіледі, атап айтқанда, квадраттардың, кубтардың және n- номдардың биквадрат формулалары. nкубтар қосындысын жіктеу үшін эффекті формуласы алынды. Бұл формуладан, жеке жағдайларда жоғары қиындығы бар кӛптеген есептерді жеңіл және оңай шығаруға болады. Бұл n-ном алгебраның теңбе- теңдіктерді қолдана арқылы жеткізіледі. Сол кезде жоғары қиындығы бар есептер қарапайым есептерді шығарғандай болады.

В статье рассматриваются максимальные обобщения формул сокращенного умножения биномов на случай умножения и возведения в степень n-членных выражений (n-номов). Приводятся лаконичные непосредственные выводы формул для начальных степеней n-номов, а именно, формулы для квадратов, кубов и биквадратов n-номов. Получена эффектная формула разложения для суммы n кубов. Из этой формулы можно получить в частных случаях формулы, позволяющие легко и просто решать многие задачи повышенной трудности. Это достигается с помощью использования тождеств алгебры n-номов. Тогда задачи повышенной трудности переходят по сложности их решения к ординарным задачам.

Maximum generalizations of the binomials short multiplication formulae for the case of multiplication and exponentiation of n elements expressions (n-nomials) are considered in the article. Compact direct development of the formulae for lowest powers, namely, the

120

formulae for quadrates, cubes and biquadrates of n-nomials is given. The glamorous formula of expansion for the sum of n cubes is obtained. In particular cases from this formula one can obtain the formulae which help easily solve many complicated problems. It is achieved with a help of applying identities of n-nomial Algebra. Then according to complexity of solving complicated problems they become ordinary problems.

It is shown that applying the apparatus of n-nomials algebra to the problems, solving which is a bit of a problem, called complicated, when using formulae of traditional (binomial) algebra, is effective. It completely confirms the prophetic words of John James Sylvester, the father of invariant theory: ―The part, in some sense, more whole: the general proposition must be proved easier than its any partial case‖. Let’s consider some simplest identities of n-nomial algebra.

Theorem 1 [3].

T

he n-nomial squared is equal to the sum of n summands squared plus the sum of their pair products doubled.

Corollary. When n5 we have a quadrate of the pentanomial:

 

     

 

.

2

2 2 2 2 2 2

de e d c e d c b e d c b a

e d c b a e d c b a









































Supposing e0, we have a quadrate of the tetranomial:



^{abcd}



^{2 a2b2c2d22}



^a



^bcd

 

^{b cd}



^cd



^.

If d e0, then we have a quadrate of the trinomial:



^abc



^{2 a2 b2 c2 2}



^a



^bc



^bc



^.

At last, if cd e0, then we have a quadrate of the binomial:



^abc



^{2 a2 b2 2ab  a2 2abb2}

Here the sign  means ―more expressive or more informative‖.

Theorem 2 [3]. The cube of the n-nomial is equal to the sum of n summands cubed plus the tripled sum of the products of one element doubled and the other one plus sextuple sum of triproducts of the elements:

  _{ }   _













 



 











k j i

k j i j

i

j i j i n

i i

n a aa a a aa a

a a

a 3 6

1 3 3 2

1  .

Corollary. The tetranomial cubed (n4) is equal to

 

           

 

   

 

.

6 3

3 3 3 3 3

d c b a d c b a

d c d c d b d b c b c b d a d a c a c a b a b a

d c b a d c b a



















































The expression in the square brackets supposes reduction:

       



^{a2 bcd b2 acd c2 abd d2 abc}



^.

If d 0, then



^abc



^{3 a3b3c33}



^a2



^bc



^b2



^ac



^c2



^ab

 

^6abc.

When d c0 we have



^ab



^{3 a3b33ab}



^ab



^{ a33a2b3ab2b3.}

Let’s emphasize that the latter (school) form is less expressive (informative) than the form



^{a b}



b a b

a³  ³ 3  containing the elementary symmetrical functions ₁

 

^{a,b ab}and

 

^{a,b ab}

2 , the latter form enables us to apply the elegant apparatus of symmetrical functions if desired.

The following original identity and its corollaries makes it easier to solve many complicated problems.

121

Theorem 3 [3]. The sum of n cubes is equal to the product of n summands added and their integrated partial quadrates of differences plus the sum of the triple products tripled:

    













 



 















k j i

k j i j

i j i n

i i n

n a a a a a a a a a

a a

a 3

1 2 2

1 3 3

2 3

1   .

Corollary. If a₁ a₂a_n 0, then















k j i

k j i

n a a a

a a

a₁³ ₂³  ³ 3 .

So, if abcd0, then ^{a3b3c3d3 3}



^ab



^cd

 

^{ ab}



^cd



, whence if d 0 , we have a³ b³ c³ d³ 3abc. Proof of this fact is considered to be a complicated problem (No 29 from [1]), though it easily derives from the corollary of Theorem 3 when taking n3.

Note. The integrated partial quadrate

 









j i

j i n

i i

n a a a

Q

1

2 is nonnegative for any a, b, c

R when n3:

 

0



2



2

2 2 2

2 2

2     ab a b  n

b a b a

Q ,

     

0



3



2

2 2

2 2

2 2

3        ab  ac  bc  n

c b c a b a c b a

Q .

But for n3, in particular, when n4



b c d

 

b c d



a d c b a

Q₄  ² ² ² ²     

is not always positive, which derived from the following statement:

Theorem 4.

  ^{ }

2 3

1

2



2



 









n

i i j

i

j i n

a n

a a

Q .

Corollary. When n2 we have

 

2

2 2 2

2

b a b

Q  a   ; if n3, then

     

2

2 2

2 3

c b c a b

Q  a     ; if n4, then

  

²

 

²

 

²

 

²



²



^{2 2 2 2}



4 a b a c a d b c b d a b c d

Q               .

We could give the formulae for the biquadrate and higher order n-nomials [3], but we will confine ourselves by theorems 1 – 4 and their consequences, which enable us to solve many complicated problems. Let’s show it using specific problems from [1, 2]:

Example 1, No29 d) from [1]: ―Factorize



^{x2  y2}

 

^{3  z2 x2}

 

^{3  y2 z2}



^3.‖

Solution: Let’s denote ax² y², b z² x² and cy² z². It is obvious that

0



b c

a , and by Theorem 3 and its corollary we have



^{2 2}



^{2 2}



^{2 2}



3 3

3 b c 3abc 3 x y x z y z

a        ,

i.e. for the apparatus of trinomial algebra (n3) this problem is trivial (comes easy);

Example 2, No29 e) from [1]: ―Factorize



^{x yz}



^{3 x3 y3 z3.‖}

Solution: The direct method of factorizing a trinomial cubed is not rational because of its bulkiness. That is why, as in Example 1, we substitute: ax yz, bx, cy and

z

d  . We have abcd0. Applying the Corollary of Theorem 3 again, we obtain

   



ab c b a b cd

 

x y



y z



x z



c b a d c b

a³ ³ ³ ³ 3 3    3    ; Example 3, No29 f) from [1]: ―Factorize x³  y³ z³ 3xyz.‖

The solution follows from Theorem 3:

122



^{x y z}

 

^{x y z xy xz yz}



Q z y x z y x z y

x³ ³ ³3    , ₃    ² ²  ²    . Example 4, No195 a) from [2]: ―Factorize



^ab

 

^{3  bc}

 

^{3  ca}



^3.‖

Solution. Suppose, xab, ybc, zca, then x yz 0, and according to Corollary of Theorem 3 we have



a b



b c



c a



z y x z y

x³ ³ ³ 3 3    .

The last example provides with the elegant generalization for arbitrary number n of cubes:

Factorize the sum of n ―cyclic differences‖ cubed



a₁ a₂

 

³ a₂ a₃



³



a ₁ a

 

³ a a₁



³

S_n      _n  _n  _n  .

Theorem 5.

       

_



 



       

 ^ _{  }

     



^{3 2 1 1 1}

1

3 2 2 1

3 a a 1 a a a a a a a a a a

S _{n n n n n}

n

i

i i i i i i

n .

The theorem is true because the sum of n cyclic differences is equal to zero, whence by Theorem 3 the conclusion of Theorem 5 follows.

Corollary.n2: S2 



ab

 

³ ba



³ ⁰, as (1,2) is not a cycle, but a transposition.

If n3, then we have Example 4:





 

 

 

 





 ^{3 3 3}

3 a b a c c a

S ³



^ab



^bc



^ca



^.

If n4, then





 

 

 

 

 

 





 ^{3 3 3 3}

4 a b b c c d d a

S ³



^ac

  

^ab



^ba

 

^{ cd}



^da

 

^{, etc.}

Thus, application of Theorem 3 provides with effective calculation of S_n for any n, though even calculation of S₃ (n3) with a help of the mechanisms of binomial algebra is considered to be a complicated problem.

Let us consider a maximum generalization of Example 2 (No29 e) from [1]):

―Factorize



^x ^y^z



³ ^x3 ^y3 ^z3.‖

We will show that even its maximum generalization



_{1 2 n}



³ ₁³ ₂³ _n³

n x x x x x x

D       for n summands is easily solved with a help of the formula of the sum of n cubes.

Theorem 6.

   











 



 











 















  











n i

i n

i

n i j

j i

n n

n x x x x x x x x x x

D

2 3

1

1 2

3 1 2

3 1   .

Proof. The terms in D_n do not contain ―pure cubes‖ ^xi³



ⁱ1,2,,ⁿ



, that is why the total number of terms in D_n is equal to n³ nL

 

n .

Let us count up the number of the terms ^R

 

ⁿ in brackets:

 

n ²



n¹



²

 

n²



n³

 

 n³



n⁴



²¹



R .

We can see that the sum in curly brackets is equal to ³₁

3

1 2

1 2

2 _





 



^{n n}

i

i C

C . Therefore,

        

3 3

6 3 1 2

1 2

3 3

1 n n

C n

n n n

n

R  ⁿ  



  







 ^ .

Multiplying ^R

 

ⁿ by the coefficient 3 we obtain that the total number of the terms in the right part is equal to

   

n n n

n n

L   

 ^{3 3}

3

3 , q. e. d.

As another application of Theorem 3 and its corollary we have the following interesting fact:

Theorem 7. The roots x₁,x₂,x₃ of the reduced cubic equation x³ pxq0 satisfy the following expression x₁³ x₂³x₃³ 3q.

123

Proof. For reduced cubic equation x₁³ x₂³ x₃³ 0 is hold. Hence using Theorem 3 we have

3 2 1 3 3 3 2 3

1 x x 3x x x

x    , but according to Viete’s Theorem we have x₁x₂x₃ q, q. e. d.

Resuming given above we can state that the formula of the sum of n cubes and its consequences considerably strengthen and complete the apparatus of binomial algebra, studied at school, bringing many so called complicated problems to the category of ordinary (not complicated) problems. Study of only two identities



^{a b c}

 

^{a b c ab ac bc}



^abc

c b

a³ ³ ³    ³ ³ ³   3 for n3 and



^{  }

 

^{   }



^{ }

 

^{ }



^



^







b c d a b c d a b c d a b c d b c d cd

a^{3 3 3 3 3 3 3 3}

   



^{ab cd  ab cd}



3 _for n4,

and then solving the problems of the kind:

1) a²b²c² abacbc (No120, a) from [1]);

2) 3



a²b²c² d²



2



a



bcd

 

b cd



cd



(of the author) and its generalization for n-nomials;

3) a⁴ b⁴ c⁴ abc



abc



(No120, d) from [1]) and its generalization by the author;

4) n



n



n n n

n a a aa a a a a

a₁^1  ₂^1  ^1  _{1 2} ₁  ₂  and others become trivial.

The author’s personal working experience (over 20 years) with schoolchildren from different specialized schools (from humanitarian to Physics and Mathematics schools, including the republican Physics and Math school) proved that pupils take a great interest when acquiring identities and inequalities of n-nomial algebra, and their scientific projects are awarded with the highest prizes at prestigious international contests (Kolmogorov and Sakharov’s reading, Euromath and others).

Objections (oral) of opponents (in particular, of some teachers) about complexity and difficulty of this subject are completely groundless. It is a result of passivity of their thinking.

On the contrary, pupils are fascinated and inflamed by the harmony and beauty of the general formulae:



₁  ₂  



² 



² 2



 ;

j

i i j

i

n a aa

a a

a 



a₁ ^a₂ ^^an



^{3 }

_

ai^{3 }3

_

_ijaiaj



ai ^aj



^5

_

_ijkaiajakand so on.

So, if n3:



^abc



^{3 a3b3c33}



^ab



^ab



^ac



^ac



^bc



^bc

 

^6abc,

which, if n2, is written as



^ab



^{3 a3 b3 3ab}



^ab



, which is more preferable and expressive than writing which is used at school:



^ab



^{3 a3 3a2b3ab2 b3, as}



^ab



^{2 a2 b2 2ab} instead of



^ab



^{2 a2 2abb2} because of collision of false symmetry harmfully for the essence of the algorithm of raising n-nomial into power. The father of Dialectics in XVII century wrote: ―That what in former eras only grown up minds of pundits studied later became available for boys’ understanding‖ (G.Hegel). Per aspera ad Veritas! (Across thorns to the truth!)or, as the father of Logics said to his great teacher:

―Amicus, Plato, sed magis amica est veritas‖ (Plato is my friend but the truth is dearer than friendship).

1. B.M.Ivlev and others. Complicated problems. M. Prosveschenie. 1990, p. 49 2. I.L.Babinskaya Math Olympiads problems. M. Prosveschenie. 1977

3. A.M.Nurlybayev Identities and inequalities of n-nomial algebra. Daryn. No 4, 2006

124 UDC 519.95

А. Nurlybaev, S. Abdykarimova

HYBRID PROGRESSIONS: SUMS FORMULA AND APPLICATIONS

(KazNPU after Abai, Almaty City)

Бұл мақалада арифметикалық және геометриялық прогрессиялардың комбинациялары қарастырылады. Олардың қосымдысылардың формулары зерттелген.

Жеке арифметикалық-геометриялық прогрессия, арифметикалық-геометриялық прогрессия және арифметикалық-геометриялық прогрессия қарастырылған. Фигурлық және пирамидалдық сандар жоғары реттік, екінші реттік және үшінші реттік прогрессиялар кұрастыруы кӛрсетілген.

В статье изучаются свойства прогрессий, являющихся комбинациями арифметической и геометрической прогрессий, а также рассмотрены их обобщение - прогрессии высших порядков. Получены точные формулы сумм их n первых членов.

Подробно (с выводом формул) исследуются арифметико-геометрическая, арифметико- арифметическая прогрессии. Показано, что последовательность фигурных (пирамидальных) чисел образует соответственно прогрессию второго (третьего) порядка, а последовательность m-тых степеней натуральных чисел образует прогрессию порядка m.

The article considers the properties of the progressions, which are combinations of arithmetical and geometric progressions. Their generalizations, i.e. higher order progressions, are considered, as well. The exact formulae for sums of their n terms are obtained. Arithmetical-geometric, arithmetical-arithmetical progressions are researched in details (with obtaining the formulae). It is shown that the sequence of figure (pyramidal) numbers forms a second (third) order progression, and the sequence of the m-th orders of natural numbers forms the m-th order progression.

Combinations of Arithmetical and Geometric Progressions (A.P. and G.P.) and their properties are studied in this paper.

1. Arithmetical Progression.

Definition. If certain quantities increase ordecrease by the same constant (d≠0) then such quantities form a series which is called an arithmetical progression.

Notation. The first term of the series is denoted by a, ( ), common difference by d, the last term by ,the number of terms by n, sum of its nterms by , and its term by .

Standard Results: ( ) a₂  ( )

( ) Note. Arithmetic Mean ( A.M.)

The arithmetic mean between two given quantities a and bis x such that a,x,b are in A.P. i.e. x-a=b-x or 2x=a+b. Hence x= (a + b )/2= A.M. ( Notation ).

The important notes.

(1). If each term of a given arithmetical progression is increased, decreased, multiplied or divided by the same non-zero quantity then the resulting series obtained will also be in A.P.

(2). Any three numbers in AP are taken asad,a,ad; any four numbers in AP are taken as ( It’s partial case of (1) when the same quantity is 2) and so on for any same quantity kd (kϵN).

Two important Properties of A.P.

125

(1) In an A.P. the sum of the terms equidistant from the beginning and end is constant and equal to the sum of the first and last terms.

(2) Any term of an A.P. ( except the first one) is equal to half the sum of the terms which are equidistant from it : (

)

2. Geometrical Progression.

Definition. A series in which each term is obtained by multiplying the previous term by the same multiple q (q≠1) is called a Geometrical Progression (G.P.). In other words, a series in which the ratio of successive terms is constant and equal q is called a G.P. This constant ratio is called a quotient and is denoted by q. The n-th term of G.P. can be written as

b_n  2.1. Sum of n terms of G.P.

( )

( ) ( ) 2.2. Sum of an infinite number of terms of a G.P.when |q|<1.

Since |q|<1 and n→∞( n tends to infinity ) then and hence in this case from (2.1)

Common ratio i.e. dividing any term by the term which precedes it.

Single geometric mean between A and B.

Suppose x is the single geometric mean between two given quantities a and b , then a,x,b are in G.P.: or √ a geometric mean between two quantities a and b.

If is a geometric means between a and b, then will be a G.P. of (n + 2) terms, in whichthe last term is b and the first term is a:

( ) ^{( )}, aq a a On putting the value of q, weshall find the n-th geometric means.

An Important Note.

1) If each term of G.P. is multiplied or divided by the same non – zero quantity h (h≠0), then the resulting series is also a G.P.

2) Odd number of terms in G.P. must be taken as ₃

q b , . . .

An even number of terms in a G.P. must be taken as

In particular three terms are taken as bq, b, b/q and four terms - as aq³,aq,a/q,a/q³. 3. Arithmetic – geometric progression.

Definition. A series in which each term g_iis the product of the corresponding terms of an A.P. and a G.P. : g_i ( ϵ ϵ ) is called an Arithmetical – Geometric Progression ( A.G.P.).

For example. ( )

In general, ( ) ( ) ( ( ) ) . Sum of n first termsof an A.G.P.

Let ( ) ( ) ( ) ].

126 Let’s calculate

Theorem 1. ^{( ) (} _{( )} ⁾].

Proof. Multiplying both sides of by the quotient q and writing as below, i. e. starting the value of by writing its first term below the 2nd term of etc.,

( ) ( ) ( ) Subtracting from , we have

( ) ( )

The middle bracket is a G.P. consisting of (n – 1) terms with its sum

( ) .

Finally. We obtain ⁽ _{( )} ⁾ .Q.e.d.

Corollary1. Putting d = 0 we have =a. Hence ( )

( ) It’s the sum of G.P. :

Corollary 2. If q = 1 (exactly, q→1 ) we have

lim

q1

, the uncertainty of the 2nd order.

Twice using L’Hospital Rule (1703) we finally havedirectly

( ) . It’s the sum of the first n terms of A.P. at b=1.

Corollary 3.If n→∞ and |q|<1 we have infinitely decreasing A.G.P. with its sum S:

[ _{( )} ]

Note. If d = 0. Wehave . It’s the sum of infinitely decreasing G.P.

Thus, A.G.P. is hybrid of A.P. and G.P.

Example 1.Calculate Solution. The terms of S_nform an A.G.P. with

a=b=d=1, =n and q=x.

( ) ( )

( ) ( )

If x=1 we have ( ) ^{( )} Example 2. Calculate ( )

Solution. We have the A.G.P. with a=b=q=3, and d=2. Substituting these means for a, b, q, and d in the formula for the sum of n terms of an A.G.P. after elementary transformation, finally, we have an elegant result:

∑ ( ) .

If we want to calculate the sum S_n ∑ ( ) we have an A.G.P. with a =1, b = q =3, and d =2. After similar calculations we obtainS_n ( ) ( ) .

Note. Interesting fact : ratio ~ 1 but its difference ~ ∞.

It’s said that one should bevery careful when operating with the notion of infinity.

Example 3. Calculate the sum .

127

Solution. Here is the A.G.P. with a=b=d=1, =n and q =1/x. Substituting these means into the formula of the sum of the A.G.P. after some elementary transformations we obtain ^



^xn1



^n1



^xn



^/



^xn1



^x1



²



^.

If x =1 this formula gives ( ) .

Example 4. Calculate the sum ( )in general at arbitrary t (tϵR).

( ) ( ) |

|

1

tn ( )( ) ( ) ( )

If t =1 then ( ) ( ), otherwise, if t =2 then ( ) ( ) and

If t=3: ( ) and so on for any t (tϵR).

Example 5. Calculate the sum

Solution. In this case we have the A.G.P. with a = b = d =1, and q =1/2 then (aftersubstituting these means into the formula of the sum of an A.G.P. and elementary transformations) we have .

If n→∞ then . Here sign ― ‖ means asymptotical equality.

The problems solved above show a power of an A.G.P. Really, AGP is a Very Important Progression(VIP)!.

Note. It’s not difficult to remember the formula for sum S_n of A.G.P.: S_n S_nS_n,

where

 

 

²

1

1 , 1

1 

 

 

 

 ^

q q q S d q

a q S a

n n

n n

n , and, finally, S_n :b



S_nS_n



.

4. Now consider the next hybrid Geometric-Geometric Progression (G.G.P.) with the current term where ϵ {·,/} i.e. asterisk means multiplication or division.

Theorem 2. The sum of the first n terms of an G.G.P. is equal to ( ) ( ) where and .

Corollary. If then ^{( )} If then we suppose that

5. Arithmetic-Arithmetical Progression (A.A.P.) is a type of series in which each term is the product of the corresponding terms of two A.P.: ( ( ) )(

( )

Theorem 3[3]. The sum of the first n terms of an A.A.P. is ( ) ( )

Corollary. If d= then ( ) ( ) If then ( )

Example. Calculate the sum ( ) Here we have the A.A.P. with a=d=1, a’=4, d’=3 hence ( )( )

( )

6. Harmonic Progression (H.P.)

128

Definition. A series of quantities is said to be in harmonic progression when their reciprocals are in arithmetical progression e.g. 1/5, 1/7, 1/9,…, and 1/a, 1/(a+d), 1/(a+2d), … is a H.P. while their reciprocals 5,7,9,…, and a, a+d, a+2d,… are in A.P.

n-th term of H.P.:

Find the n-th term of the corresponding A.P. and then its reciprocals. If 1/a, 1/(a+d),1/(a+2d),… is a H.P., then corresponding A.P. is a, a+d, a+2d, … of A.P. is a+(n- 1)d, of H.P. is 1/[a+(n-1)d]. In order to form a H.P., we should form the corresponding A.P. first.

Harmonic mean (Single)

The harmonic mean (H.M.) between two quantities a and b is x if a, x, b are in a harmonic progression.If 1/a, 1/x, 1/b is an A.P. then and

Note. If a=d=1 in a H.P. we have a harmonic number ∑

The letter H stands for ―harmonic‖; is a harmonic number, it is called so because a tone of wave length 1/k is called the k-th harmonic of a tone whose wave length is one.

L.Euler proved that ( )= γ where γ=0,5772156649… which is now known as Euler’s constant and conventionally denoted by the Greek small letter γ(gamma).

Although the sum of harmonic numbers approaches infinity, they approach it only logarithmically – that’s, quite slowly.

7. High Order Arithmetic Progression (HOAP)

Definition (by induction): Zero order (k=0) A.P. is constant series (sequence) k=0, a,a,…,a where aϵ R, then a first order A.P. is such series in which differences

form ZOAP(Zero Order A.P.) and so on, the series form k- th order A.P.

(AP-k) if differences form (k-1) -th order A.P.

Example. The sequence 1, 3, 6, 10, 15, 21,…, is 2^nd order A.P. because its differences form the first order AP (A.P.-1).

Indeed.

AP-2: 1 3 6 10 15 21 28 36 45 55 66 78 … AP-1: 2 3 4 5 6 7 8 10 11 12 … n

AP-0: 1 1 1 1 1 1 1 1 1 1 1 Note. is atriangle number.

Theorem 4. The series of figure (plane) numbers ( ) form the 2ndorder A.P., where ( ) .

Theorem 5. The series of pyramidal (space) numbers ( ) form the 3d order A.P.

Theorem 6 [3].The partial sums of figure (pyramidal) numbers form the third (fourth) order A.P.

Theorem 7 [3]. The m powers of natural numbers 1^m,2^m,3^m,, n^m form the m-th order A.P.

Theorem 8. The sum Sm

 

n 1^m2^m n^m is a rational polynomial of power

1

m of number n, i.e. Sm

 

n ^O

 

n^m1 . Theorem 9 [3]. For initial values of m:

1

m : S1

  

n n n1



/2O

 

n² ; m2: S2

 

n S1

 

n 2n1



/3O

 

n³ ;

3

m : S3

 

n 



S1

 

n



²O

 

n⁴ ;m4: S4

 

n S2

 

n



3n²3n1



/5O

 

n⁵ ;

129

5

m : S5

 

n S3

 

n



2n² 2n1



/3O

 

n⁶ ;

6

m : S6

 

n S2

 

n



3n⁴ 6n³ 3n1



/7O

 

n⁷ ;

7

m : S7

 

n S3

 

n



3n⁴6n³n²4n2



/6O

 

n⁸ ;

8

m : S8

 

n S2

 

n



5n⁶15n⁵5n⁴15n³2n²9n3



/5O

 

n⁹ ;

9

m : S9

 

n S3

 

n



2n⁶6n⁵n⁴8n³n²6n3



/5O

 

n¹⁰ ;

10 m :

 

2

  

^{8 7 6 5 4 3 2}

  

¹¹

10 n S n 3n 12n 8n 18n 10n 24n 2n 15n 5 /11 O n

S           ;

And so on, ^Sm

 

ⁿ can be directly calculated using the identity



1



1 1

1    

C _ C C_^ k n C_k^k _k^k  _n^k _n^k .

For example,

 

²₁

1 1 1

1 











 

^{n n}

i i n

i

C C i

n

S ,

  

2



2 ³₁ ²₁

 

1



2 1

 

/6

1 2 1

2

2     _  _   









^{i C i C C n n n}

n

S _{n n}

n i

i n

i

     

²1 ²

2 1 4

2 1

3 1 1

3

3 6 6 _{  }

 















 

^{n n n n}

i i n

i

C C

C i C i

n

S and so on.

1. Wander Warden. Algebra. M.Nauka. 1976

2. R.Graham, D.Knuth, O.Patasnik, Concrete Maths. Addison-Wesley. 2^nd ed., 1998 3. Nurlybaev A. Hybrid progressions: Sums Formula and Applications // Fizmat №3, 2012

ӘОЖ 532.13

Х.Ш. Таирова, Ғ.Д. Балымбет

ЖАНАР МАЙЛАРДЫҢ ФИЗИКАЛЫҚ ҚАСИЕТТЕРІН ЗЕРТТЕУ

(Алматы қ., № 59 мектеп-гимназия)

Автомобиль отынының сапасын анықтайтын – октандық сан, тығыздық, тұтқырлық, беттік керілу және булану сияқты жанар майдың физикалық қасиеттері қарастырылған. Қатты дененің тұтқыр ортадағы қозғалыс заңы келтірілген және зертханалық жағдайда Стокс әдісімен тұтқырықты анықтаудың есептеу формуласы қорытылып шығарылған. Тәжірибе параметрлерін тағайындау үшін глицириннің тұтқырлығы анықталған және кестелік мәнмен салыстырылды. Стокс әдісімен маркалары әр түрлі жанар майлардың тұтқырлығын анықтаудың зерттеу нәтижелері келтірілген.

Рассматривается физические свойства бензина – как октановое число, плотность, вязкость, поверхностное натяжение и испаряемость, определяющий качество автомобильного топлива. Приводится основные законы движения твердого тела в вязкой среде и выводится расчетная формула для определения вязкости методом Стокса в лабораторных условиях. Для установления параметров эксперимента определена вязкость глицирина и сравнивается с табличными данными. Приводится результаты исследования определения вязкости различных марок бензина методом Стокса.

Regarded the physical properties of gasoline - as octane number, density, viscosity, surface tension and evaporability that determines the quality of motor fuel. Contained the

130

basic laws of motion of a solid body in a viscous medium and excreted design formula for determining the viscosity the Stokes method in the laboratory. To establish the experimental parameters is defined viscosity glitsirina and compared with reference data. Contained results of a study to determine the viscosity of various grades of gasoline by Stokes.

Бүкіл әлемде автомобиль отыны – жанар майдың кӛптеген түрлері ӛндіріледі және тұтынады. Жанар май автомобиль цилиндрінде «дұрыс» жануы үшін ол бірнеше қасиеттерге ие болуы керек. Оның маңызды қасиетінің бірі – октандық сан. Барлық май құю бекетінде осы октандық сан жазылған және жанар майдың сапасы мен құны осы санға байланысты.

Мұнайдың ӛнімі болып табылатын жанар майдың сапасын анықтау барысында тұтқырлық негізгі кӛрсеткіш болып табылады және осы арқылы мұнайға баға беріледі.

Тұтқырлық мұнай ӛнімдерінің қозғалу дәрежесін және оның тығыздығы мен құрамына қарай ағу жылдамдығын анықтайды. Осыған орай май тұтқырлығы механизмдер мен қозғалтқыштардың үйкеліс шамасын, дәлірек айтсақ, үйкеліске шығындалатын энергияның шамасын анықтайды.

Жанар майлар тұтқырлығы тӛмен мұнай ӛнімдеріне жатады. Олардың механикалық құрамы, заңды түрде, пайдалану кезінде қиындықтар туғызбайды және соның салдарынан бұл мәселеге тұтынушылар мен технологтар кӛп кӛңіл аудармайды.

Сол себепті заманауи автомобильдер тұтынып жүрген жанар майлардың тұтқырлығын зерттеуді алға мақсат етіп қойдық. Жазғы және қысқы деп бӛлінетін мотор майларының тұтқырлығын және температураға тәуелділігін де зерттеуге болады. Бірақ, ол болашақтың мәселесі.

Сұйықтардың тұтқырлығын немесе ішкі үйкеліс коэффициентін тәжірибе жүзінде Стокс немесе ротациялық вискозиметр әдістерін қолданып анықтауға болады.

Стокс әдісі тұтқырлығы тӛмен мысалы, глицерин, жанар май сияқты сұйықтардың ішкі үйкеліс коэффициентін анықтауға қолданылады. Тұтқырлықты анықтаудың бұл әдісі, сұйық ішінде баяу қозғалатын сфера пішінді кішкене қатты дене – шардың жылдамдығын ӛлшеуге негізделген.

Дене тұтқыр ортада қозғалғанда пайда болатын кедергінің екі түрлі себебі бар.

1) Жылдамдығы аз, дененің пішіні сусымалы болған жағдайда кедергі күші тек сұйықтың тұтқырлығынан пайда болады.

2) Қатты денеге тікелей тиісіп жатқан сұйық қабаты оның бетіне жабысады және сол денеге ілесіп қозғалады. Келесі қабат денеге ілесіп аз ғана жылдамдықпен қозғалады. Сӛйтіп, сұйық қабаттарының арасында үйкеліс күші пайда болады.

Бұл жағдайда Стокс тағайындаған заң бойынша: кедергі күші ортаның тұтқырлық коэффициентіне, дененің сызықтық өлшеміне және дененің қозғалыс жылдамдығына тура пропорционал болады. Сонда тұтқыр сұйық ішінде қозғалған шарға Стокс заңы бойынша пайда болатын кедергі күші:



r

F 6 . (1) Тұтқыр сұйық ішінде еркін құлаған шарға P

ауырлық күші, Q

Архимедтің кері итеруші күші және F

тұтқырлықтың кедергі күші (қозғалысқа кедергі күші) әсер етеді.Тұтқыр сұйықта құлаған шардыңқозғалысы тек алғашқы уақыттарда үдемелі болады. Жылдамдық артқан сайын тұтқырлықтың кедергі күші артады және қандай да бір уақыт моментінде қозғалысты бірқалыпты деп есептеуге болады, яғни тӛмендегі теңдік орындалады

0



F Q P  

. (2) Қорыта келе, алатынымыз

In document Просмотр «Том 40 № 4 (2012): Вестник КазНПУ им.Абая. Серия: физико-математические науки» | «Физико-математические науки» (бет 120-145)