If in the expansion of $(1+\text{y})^{\text{n}},$ the coefficients of 5 th , 6 th and 7 th terms are in A.P., then n is equal to:

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Aicharya · Accepted Answer

Solution: Coefficients of the 5 th , 6 th and 7 th terms in the given expansion are ${^\text{n}}\text{C}_{\text{4}},{^\text{n}}\text{C}_{\text{5}}$ and ${^\text{n}}\text{C}_{\text{6}}.$ These coefficients are in AP. Thus, we have $2\ {^\text{n}}\text{C}_{\text{5}}={^\text{n}}\text{C}_{\text{4}}+{^\text{n}}\text{C}_{\text{6}}$ On dividing both sides by ${^\text{n}}\text{C}_{\text{5}},$ we get $2=\frac{{^\text{n}}\text{C}_{\text{4}}}{{^\text{n}}\text{C}_{\text{5}}}+\frac{{^\text{n}}\text{C}_{\text{6}}}{{^\text{n}}\text{C}_{\text{5}}}$ $\Rightarrow 2=\frac{5}{\text{n}-4}+\frac{\text{n}-5}{6}$ $\Rightarrow 12\text{n}-48=30+\text{n}^{2}-4\text{n}-5\text{n}+20$ $\Rightarrow \text{n}^{2}-21\text{n}+98=0$ $\Rightarrow (\text{n}-14)(\text{n}-7)=0$ $\Rightarrow \text{n}=7,14$

If in the expansion of \((1+\text{y})^{\text{n}},\) the coefficients of 5 th , 6 th and 7 th terms are in A…

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