If the coefficients of 5th, 6th , and 7th terms in the expansion of `(1+x)^n` are in A.P., then `n=` a. 7 only b. 14 only c. 7 or 14 d. none of these

Explanation

Coefficients of the 5 th, 6th and 7 th terms in the given expansion are `\ ^{n} C_{4}\ ^{n} C_{5}` and `\ ^{n} C_{6}`

These coefficients are in AP.

Thus, we have


`2^{n} C_{5}=\ ^{n} C_{4}+\ ^{n} C_{6}`


On dividing both sides by `\ ^{n} C_{5}`, we get:

`2=\frac{\ ^{n} C_{4}}{\ ^{n} C_{5}}+\frac{\ ^{n} C_{6}}{\ ^{n} C_{5}}` `\Rightarrow 2=\frac{5}{n-4}+\frac{n-5}{6}`

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