Binomial Theorem 1 Question 26
28. Let $n$ be a positive integer. If the coefficients of $2 nd, 3 rd$, and 4th terms in the expansion of $(1+x)^{n}$ are in AP, then the value of $n$ is… .
(1994, 2M)
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Solution:
- Let the coefficients of $2 nd, 3 rd$ and 4 th terms in the expansion of $(1+x)^{n}$ is ${ }^{n} C _1,{ }^{n} C _2,{ }^{n} C _3$.
According to given condition,
$$ \begin{array}{rlrl} & & 2\left({ }^{n} C _2\right) & ={ }^{n} C _1+{ }^{n} C _3 \\ \Rightarrow & 2 \frac{n(n-1)}{1 \cdot 2} & =n+\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3} \\ \Rightarrow & n-1 & =1+\frac{(n-1)(n-2)}{6} \\ \Rightarrow & n-1 & =1+\frac{n^{2}-3 n+2}{6} \\ \Rightarrow & & 6 n-6 & =6+n^{2}-3 n+2 \\ \Rightarrow & & n^{2}-9 n+14 & =0 \\ \Rightarrow & & (n-2)(n-7) & =0 \\ \Rightarrow & n & n \\ \text { or } & & n & =7 \end{array} $$
But ${ }^{n} C _3$ is true for $n \geq 3$, therefore $n=7$ is the answer.
