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Question

Let n be a positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of (1+x)n are in AP, then the value of n is ___ .


Solution

The coefficients of 2nd, 3rd and 4th terms in the expansion of (1+x)n is nC1, nC2, nC3
According to the given condition,
2(nC2)=nC1+nC3   2n(n1)1.2=n+n(n1)(n2)1.2.3   n1=1+(n1)(n2)6   n1=1+n23n+26   6n6=6+n23n+2   n29n+14=0   (n2)(n7)=0   n=2,7

But nC3 is true for n 3, therefore n = 7 is the answer.

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