Home
Class 12
MATHS
If S(1), S(2), S(3),...,S(n) are the sum...

If `S_(1), S_(2), S_(3),...,S_(n)` are the sums of infinite geometric series, whose first terms are 1, 2, 3,.., n and whose common rations are `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n+1)` respectively, then find the values of `S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)`.

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(6) (2n) (2n +1) (4n +1) -1`
Consider an infinite GP with first term 1,2,3,.., n and common ratios `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n +1)`
`:. S_(1) = (1)/(1 -1//2) = 2`
`{:(" "S_(2) = (2)/(1 -1//3) = 3),(" "vdots " "vdots " "vdots),(S_(2n -1) = (2n -1)/(1-1//2n) = 2n):}`
`:. S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)`
`= 2^(2) + 3^(2) + 4^(2) + ...+ (2n)^(2)`
`= (1)/(6) (2n) (2n+1) (4n+1) -1`
Promotional Banner

Similar Questions

Explore conceptually related problems

If S_(1), S_(2), S_(3), …, S_(m) are the sums of n terms of m A.P.'s whose first terms are 1, 2, 4, …, m and whose common differences are 1, 3, 5, …, (2m-1) repectively, then show that S_(1)+S_(2)+S_(3)+…+S_(n)=(1)/(2)mn(mn+1)

Let S_(k) , where k = 1,2 ,....,100, denotes the sum of the infinite geometric series whose first term is (k -1)/(k!) and the common ratio is (1)/(k) . Then, the value of (100^(2))/(100!) +sum_(k=2)^(100) | (k^(2) - 3k +1) S_(k)| is....

If S_k=(1+2+3+...k)/k then find the value of S_1^2+S_2^2+....S_n^2

If S_1, S_2, S_3 are the sum of first n natural numbers, their squares and their cubes, respectively , show that 9 S_(2)^(2) = S_(3) (1+ 8S_1)

If S_(n) denotes the sum of n terms of an A.P., S_(n +3) - 3S_(n + 2) + 3S_(n + 1) - S_(n) =

In a geometric series , the first term is a and common ratio is r. If S_n denotes the sum of the n terms and U_n=Sigma_(n=1)^(n) S_n , then rS_n+(1-r)U_n equals

Let sum of n , 2n , 3n , terms of an A.P are S_(1), S_(2), S_(3) respectively. Prove that S_(3) = 3 (S_(2) - S_(1)) .

If S_(n)=(1.2)/(3!)+(2.2^(2))/(4!)+(3.2^(2))/(5!)+...+ up to n terms, then sum of infinite terms is