Home
Class 12
MATHS
Find the sum of series upto n terms ((2n...

Find the sum of series upto n terms `((2n+1)/(2n-1))+3((2n+1)/(2n-1))^2+5((2n+1)/(2n-1))^3+...`

Text Solution

Verified by Experts

For `x != 1`, let
`S = x + 3x^(2) + 5x^(3) + ... + (2n - 3) x^(n - 1) + (2n - 1) x^(n)`...(i)
`rArr xS = x^(2) + 3x^(3) + ...+ (2n - 5) x^(n - 1) + (2n - 3) x^(n) + (2n - 1) x^(n + 1)`...(ii)
Subtracting (ii) from (i), we get
`(1 - x) S = x + 2x^(2) + 2x^(3) + ... + 2x^(n - 1) + 2x^(n) - (2n - 1) x^(n + 1) = x + (2x^(2) (1 - x^(n - 1)))/(1 - x) - (2n - 1) x^(n + 1)`
`=(x)/(1 - x) [1 - x + 2x - 2x^(n) - (2n -1) x^(n) + (2n - 1) x^(n + 1)]`
`rArr S = (x)/((1 - x)^(2)) [(2n - 1) x^(n + 1) - (2n + 1) x^(n) + 1 + x]`
Thus `((2n + 1)/(2n - 1)) + 3((2n + 1)/(2n - 1))^(2) + ....+ (2n -1) ((2n + 1)/(2n -1))^(n)`
`= ((2n + 1)/(2n -1))((2n - 1)/(2))^(2) [(2n - 1) ((2n + 1)/(2n -1))^(n+ 1) - (2n + 1) ((2n + 1)/(2n -1))^(n) + 1 + (2n + 1)/(2n - 1)]`
`= (2n^(2) - 1)/(4) .(4n)/(2n -1) = n (2n + 1)`
Promotional Banner

Topper's Solved these Questions

  • SEQUENCE AND PROGRESSION

    ALLEN|Exercise Do yourself|3 Videos

  • SEQUENCE AND PROGRESSION

    ALLEN|Exercise Do yourself 2|2 Videos

  • RACE

    ALLEN|Exercise Race 21|10 Videos

  • TEST PAPER

    ALLEN|Exercise CHEMISTRY SECTION-II|8 Videos

Similar Questions

Explore conceptually related problems

find the sum of the series upto n terms 1+(1+2)/(2)+(1+2+3)/(3)+(1+2+3+4)/(4)+

Find sum of the series upto n terms 1+(1+2)/(2)+(1+2+3)/(3)+(1+2+3+4)/(4)+

[ If 2 is the sum of infinity of a G.P.,whose first clement is 1 ,then the sum of the first n terms is [ 1) (2^(n)-1)/(2^(n)), 2) (2^(n)-1)/(2^(n-1)), 3) (2^(n-1)-2)/(2), 4) (2^(n-1)-1)/(2^(n))]]

find the sum of n terms of the series (4-(1)/(n))+(4-(2)/(n))+(4-(3)/(n))+

Find the sum of the series 1xx n+2(n-1)+3xx(n-2)+...+(n-1)xx2+n xx1

Find the sum of n terms of the series (1)/(2*4)+(1)/(4*6)+... (A) (n)/(n+1) (B) (n)/(4(n+1)) (C) (1)/((2n)(2n+2))( D) )(1)/(2^(n)(2^(n)+2))

The sum of the series 1^2+3^2+5^2+ddotto\ n terms is a. (n(n+1)(2n+1))/2 b. (n(2n-1)(2n+1))/3 c. ((n-1)^2(2n+1))/6 d. ((2n+1)^3)/3

Find the sum of the series: 1.n+2.(n-1)+3.(n-2)+...+(n-1).2+n.1

Find the sum of the following series to n term: (n^2-1^2)+2(n^2-2^2)+3(n^2-3^2)+……...

Find the sum of the following series whose n t h term is: 2n^3+3n^2-1.