SATHEE: Binomial Theorem 1 Question 26

Binomial Theorem 1 Question 26

28. Let $n$ be a positive integer. If the coefficients of $2 n d, 3 r d$ , 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 $2 n d, 3 r d$ 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,

$\begin{aligned} 2 (^{n} C_{2}) & =^{n} C_{1} +^{n} C_{3} \\ \Rightarrow & 2 \frac{n (n - 1)}{1 \cdot 2} & = n + \frac{n (n - 1) (n - 2)}{1 \cdot 2 \cdot 3} \\ \Rightarrow & n - 1 & = 1 + \frac{(n - 1) (n - 2)}{6} \\ \Rightarrow & n - 1 & = 1 + \frac{n^{2} - 3 n + 2}{6} \\ \Rightarrow & 6 n - 6 & = 6 + n^{2} - 3 n + 2 \\ \Rightarrow & n^{2} - 9 n + 14 & = 0 \\ \Rightarrow & (n - 2) (n - 7) & = 0 \\ \Rightarrow & n & n \\ or & n & = 7 \end{aligned}$

But $^{n} C_{3}$ is true for $n \geq 3$ , therefore $n = 7$ is the answer.

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Binomial Theorem 1 Question 26

28. Let n be a positive integer. If the coefficients of 2nd,3rd, and 4th terms in the expansion of (1+x)n are in AP, then the value of n is… .

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28. Let $n$ be a positive integer. If the coefficients of $2 n d, 3 r d$ , and 4th terms in the expansion of $(1 + x)^{n}$ are in AP, then the value of $n$ is… .