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

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

Text Solution

Verified by Experts

Since n is an odd integer, `(-1)^(n-1)=1` and n-1,n-3,n-5,… are even integers.
The given series is
`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)]`
`=(n^(2)(n+1)^(2))/4-2xx2^(3)[1^(3)+2^(3)+3^(3)+..+((n-1)/2)^(3)]`
`=(n^(2)(n+1)^(2))/4-16[1/2((n-1)/2)((n-1)/2+1)]^(2)`
`=(n^(2)(n+1)^(2))/4-((n-1)^(2)(n+1)^(2))/4`
`=((n+1)^(2))/4[n^(2)-(n-1)^(2)]`
`=1/4(n+1)^(2)(2n-1)`
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