Home
Class 12
MATHS
For any odd integer nge1, find the sum n...

For any odd integer `nge1`, find the sum `n^(3)-(n-1)^(3)`+…..+`(-1)^(n-1)1`.

Text Solution

Verified by Experts

Since, n is an odd interger, `(-1)^(n-1)=1`
`and n-1,n-3,n-5` etc., are even intergers, then `n^(3)-(n-1)^(3)+(n-2)^(3)-(n-3)^(3)+ . . .+(-1)^(n-1).1^3`
`=n^(3)+(n-1)^(3)+(n-2)^(3)+ . .+1^(3)`
`-2[(n-1)^(3)+(n-3)^(3)+ . . .+2^(3)]`
`=sumn^(3)-2xx2^(3)[((n-1)/(2))^(3)+((n-3)/(2))^(3)+ . . .+1^(3)]`
`[beausen-1,n-3` . . ., are even intergers]
`=sumn^(3)-16[sum((n-1)/(2))^(3)]`
`=[(n(n+1)/(2)]^(2)-16[(1)/(2)((n-1)/(2))((n-1)/(2)+1)]^(2)`
`=(1)/(4)(n+1)^(2)-(16(n-1)^(2)(n+1)^(2))/(4xx4xx4)`
`=(1)/(4)(n+1)^(2)[n^(2)-(n-1)^(2)]=(1)/(4)(n+1)^(2)(2n-1)`
Promotional Banner

Similar Questions

Explore conceptually related problems

For and odd integer n ge 1, n^(3) - (n - 1)^(3) + …… + (- 1)^(n-1) 1^(3)

Statement 1: Sum of the series 1^3-2^3+3^3-4^3++11^3=378. - Statement 2: For any odd integer ngeq1,n^3-(n-1)^3++(-1)^(n-1)1^3=1/4(2n-1)(n+1)^2dot

Find the sum Sigma_(r=1)^(oo)(3n^2+1)/((n^2-1)^3)

Find the sum Sigma_(j=1)^(n) Sigma_(i=1)^(n) I xx 3^j

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .

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

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .