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The construction of the sums of a geometric progression to a degree
Section 1. Mathematics
Druzhinin Victor Vladimirovich, Alekseev Vladimir Vasilevich National research University "MEPHI" Sarov physical-technical Institute E mail: vvdr@newmail.ru
The construction of the sums of a geometric progression to a degree
Abstract: The analytical expressions for the erection of sums of geometric progressions to a degree. The results are used for the summation of series, complex finite sums, solving algebraic equations. Keywords: geometric progression, complex finite and infinite sums.
In analytical mathematics there are many examples of finite sums and series, with a relatively simple formula the result. These include the binomial theorem, sums of arithmetic and geometric progressions, sums of Bernoulli — ordered amount ofthe one power, whole and fractional numbers, arithmetic-geometric progressions and other subjects [1; 2; 3; 4]. In this article we focus on the construction in the natural degree m amounts of a geometric progression
G(a ,n ,m) =
=|S»'I =(1
L + a + a + a +... + a
„«+1 i \ a -1
a-1
(1)
In reference books and educational literature we have not found this problem. Meanwhile here there is a large set of new deemed according to the formula amounts and ranks as well as taking the integrals and the solution of algebraic equations. In (1) the final result is the right side, but as reveals the amount of degree, what its symmetry and structure are we show below. First we write the polynomial theorem
2X I =X<
m !
-U ) (x2 )\..(xn )\ (2)
(I!)(f 2!)... (.!)
where (t 1 +12 + ... + tn) = m and the sum goes through all combinations of these numbers. With the special structure of numbers xk = xk (our case) the left part is much easier. Number of terms is (nm +1) ,the highest degree is anm. General record in nm even has a view
G (a ,n,m) = £ckak + £ bkak + dsas + "gV + £ ffi. (3)
k=0 k=n+1 k=s+1 k=2s—n
Coefficients with increasing k monotonically increase up to including ds. The central coefficient ds has the greatest value. The following ratios monotonically
decrease symmetrically, and ck = fnm-k and bk = enm-k. We have identified the coefficients of the different letters as they are considered by different formulas. The basic factors are calculated using binomial coefficient (n > 2)
ck (n,m) =
(m + k -1)!
=ct
(4)
(m - l)!k!
For the coefficients bk and dswe have not received General formula, but found them to specific m = 2,3,4. If nm is an odd number, then in the middle of the right row (3) there is not one greatest factor, and two of the same factor. If n = 2t + l,m = 2p +1 that are coefficients d2 pt+t+p = d2 ft+t+p+1. Subsequently the coefficients of mirror symmetry occurs.
If m = 2 then
G(,n,2) = f]Tflk^ =Z( +k + X (2n +1 -k)a. (5)
V k=0 J k=0 k=n+1
For example,
(l + a + a2 )2 = 1 + 2a + 3a2 + 2a3 (l + a + a2 + a3 )2 = 1 + 2a + 3a2 + 4a3 + 3a4
From identities (6 -7) a symmetry of the right and left parts and an algorithm for the construction of any geometric progression is in the square.
In the construction of G(a, n, 3) in the cube we obtain the following result. If n is even, n = 2t, if 2t +1 < k < 3t -1 ,bk = 1 + 3t(t + l)-(mt -k)2; d3t = 1 + 3t (t +1). Examples
G (a, 2,3) = 1 + 3a + 6a2 + la3 + 6a4 + 3a5 + a6 G (a ,4,3) = 1 + 3a + 6a2 + 10a3 + 15a4 +
+18a5 + 19a6 + 18a7 +... + a12. G (a, 6,3) = 1 +... 21a5 + 28a6 + 33a7 +
+36a8 + 37 a9 + 36a10... + a18.
-a , I-2a5
(6) (7)
(8) (9)
(10)
m
m
Section 1. Mathematics
(11) (12)
If n is odd, n = 2t +1, then 2t + 2 < k < 3t,b3t+1-q = = d3t+1 - q (q +1); 1 ^ q ^ t -1. The highest coefficients
d3t+1 = d3t+2 = 3(( +1)2. Examples
G (a, 3,3) = 1 + 3a + 6a2 +10a3 +12a4 +
+12a5 + 10a6 + 6a7 + 3a8 + a ".
G (a ,5,3) = 1 + 3a + 6a2 + 10a3 + 15a4 +
+21a5 + 25a6 + 27a7 +... + a18.
These amounts are calculated according to right side (1). In the construction of G (a ,n,4) in the fourth degree we get such formulas, regardless of the evenness or unevenness of n. If k = 2n ,d2n = l + 5n + + 2n(n-l)(n + 4)/3 , the maximum value of the coefficient. If k = 2n -1 have the greatest b2n-1 = d2n _(n +1) . The other coefficients b2n l, with 1 < l < n - 2 are calculated by recurrent relations b2n-1-1 = b2n-1 - (21 + l)n + (312 -1 - 2)12. Examples
G (a ,2,4 ) = 1 + 4a + 10a2 + 16a3 +
+19a4 + 16a5 +... + a8.
G (a,3,4) = 1 + 4a + 10a2 + 20a3 + 31a4 +
+40a5 + 44a6 + 40a7 +.. + a12.
G (a, 4,4 ) = 1 +... + 35a4 + 52a5 + 68a6 +
(13)
(14)
(15)
+ 80a7 + 85a8 + 80a9... + a16. It is also possible to obtain other degrees G (a ,n ,m). Consider the possible applications of these structures.
1. The sum of a convergent series of generalized geometric progression with 0 < |a| < 1 species arises
1 (16)
+ k -1)! ak =.
(m - l)!k! (1 - a) '
2. If we take the derivative at a, we get the new calculatedamount. For example, dG (a, 2,3)/da gives
1 + 4a + 7 a2 + 8a3 + 5a4 + 2a3 =
_(a3 -1)2 (2a2 - a -1)
(a -1)3
(17)
3. There appears the possibility ofanalytical solutions of algebraic equations of high order. The equation a6 + 3a5 + 6a4 + 7a3 + 6a2 + 3a - 26 = 0 is reduced to the equation a2 + a - 2 = 0 which has two aj = 1, a2 = -2.
4. Combinations of sums and products of G (a, n ,m ) form a new hope of final amount and series and also give the opportunity to find the integral analytically.
References:
1. Sizyi S. V. Lectures on the theory of numbers, FIZMATGIZ, - M., 2007.
2. Korn. G., Korn. T. Handbook of mathematics, Science, GHML, - M., 1974. P. 31, 135.
3. Gradshteyn I. S., Ryzhik I. - M. Tables of integrals, sums, series and products. GIFML, - M, 1962. P. 15-16.
4. Druzhinin V. V., Sirotkina A. G. NTVP, No. 4, 2016, P. 15-16.
