Home
Class 12
MATHS
Find the sum of 1/(1!(n-1)!)+1/(3!(n-3)!...

Find the sum of `1/(1!(n-1)!)+1/(3!(n-3)!)+1/(5!(n-5)!)+ ,`

Text Solution

Verified by Experts

`because 1 ! = 1 `
` therefore ` The given series can be written as
`(1)/(n!)(1)/((n-1)!1!) + (1!)/((n-3)!3!) + (1)/((n-5)!5!) + ...` …(i)
` because ` Sum of values of each terms in factorial are equal
.` (n-1) + 1 = (n-3) + 3= (n-5) + 5 = … = n `
From Eq. (i)
`(1)/(n!)[(1)/((n-1)!1!) + (1!)/((n-3)!3!) + (1)/((n-5)!5!)+ ...]`
` = (1)/(n!) ( ""^(n)C_(1) + ""^(n)C_(3) + ""^(n)C_(3) + ""^(n)C_(5) + ...) = (2^(n-1))/(n!)` .
Promotional Banner

Similar Questions

Explore conceptually related problems

Find the sum of the series 1*n+2*(n-1)+3*(n-2)+4*(n-3)+....(n−1).2+n.1"

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

Find the sum of the series : (tan^- 1)1/3+(tan^- 1)2/9+....+(tan^- 1)(2^(n-1))/(1+2^(2n-1))+...... oo

Find the sum of the series (1^(2)+1)1!+(2^(2)+1)2!+(3^(2)+1)3!+ . .+(n^(2)+1)n! .

Find the sum: 1^(1) -2^(2) + 3^(2)-4^(2)+ 5^(2)- 6^(2)+ ……up to n terms

If ,Z_1,Z_2,Z_3,........Z_(n-1) are n^(th) roots of unity then the value of 1/(3-Z_1)+1/(3-Z_2)+..........+1/(3-Z_(n-1)) is equal to

Find the value of n such that (""^(n)P_(4))/(""^(n-1)P_(4))=(5)/(3), n gt 4

If n is an odd integer greater than or equal to 1, then the value of n^3 - (n-1)^3 + (n-2)^3 - (n-3)^3 + .... + (-1)^(n-1) 1^3

Find the sum to n terms cot^(-1)(3)+cot^(-1)(7)+cot^(-1)(13)+ cot^(-1) (21)+.......

Show that the middle term in the expansion of (1+x)^2n is 1.3.5…(2n-1)/n! 2n.x^n , where n is a positive integer.