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Multinomial Expansion

In the binomi...

Question

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

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Solution

Coefficients of the 5th, 6th and 7th terms in the given expansion are nC4 , nC5 and nC6

These coefficients are in AP.

Thus, we have 2 nC5 = nC4+nC6

On dividing both sides by C5n,

we get:

2=nC4/nC5 + nC6/nC5

⇒2=5/(n-4)+(n-5)/6

⇒2=56/[6(n-4)] + (n-5)(n-4) / [6*(n-4)]

⇒12n-48 = 30+n²-4n-5n+20

⇒n²-21n+98=0

⇒(n-14)(n-7)=0

⇒n=7 and 14

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