In the binomial expansion of `(1+x)^n` , coefficients of the fifth, sixth and seventh terms are in A.P. find all the values of `n` for which this can happen.

Explanation

Coefficients of fifth, sixth and seventh terms in `(1+x)^n` are `C(n,4),C(n,5) and C(n,6).`

As, they are in `A.P.`,

`:. 2**C(n,5) = C(n,4)+C(n,6)`

`=>2*(n!)/(5!(n-5!)) = (n!)/(4!(n-4!)) + (n!)/(6!(n-6!))`

`=>2*1/(5(n-5)) = 1/((n-4)(n-5)) + 1/(6*5)`

`=>2/(5(n-5)) = 1/((n-4)(n-5)) + 1/30`

`=>2/(5(n-5)) = (30+(n-4)(n-5))/(30(n-4)(n-5))`

`=>12(n-4) = 30+(n^2+20-9n)`

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