If in the expansion of $(\text{a}+\text{b})^{\text{n}}$ and $(\text{a}+\text{b})^{\text{n}}+3,$ the ratio of the coefficients of coefficients of second and third terms, and third and fourth terms respectively are equal, then n is:

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Aicharya · Accepted Answer

Solution: Coefficients of the 2nd and 3rd terms in $\Big(\text{a}+\text{b}\Big)^{\text{n}}$ are ${^\text{n}}\text{C}_{\text{}1}$ and ${^\text{n}}\text{C}_{\text{}2}$ Coefficients of the 2nd and 3rd terms in $\Big(\text{a}+\text{b}\Big)^{\text{n}+3}$ are ${^\text{n+3}}\text{C}_{\text{}2}$ and ${^\text{n+3}}\text{C}_{\text{}3}$ Thus, we have $\frac{{^\text{n}}\text{C}_{\text{}1}}{{^\text{n}}\text{C}_{\text{}2}}=\frac{{^\text{n+3}}\text{C}_{\text{}1}}{{^\text{n+3}}\text{C}_{\text{}3}}$ $\Rightarrow \frac{2}{\text{n}-1}=\frac{3}{\text{n}+1}$ $\Rightarrow 2\text{n}+2=3\text{n}-3$ $\Rightarrow \text{n}=5$

If in the expansion of \((\text{a}+\text{b})^{\text{n}}\) and \((\text{a}+\text{b})^{\text{n}}+3,\) the…

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