Question:
Consider the expansion of \((1+x)^{2n+1}\). If the coefficients of \(x^r\) and \(x^{r+1}\) are equal in the expansion, then \(r\) is equal to:
\((1+x)^{2n+1}\) के विस्तार पर विचार करें। यदि विस्तार में \(x^r\) और \(x^{r+1}\) के गुणांक बराबर हैं, तो \(r\) का मान है:
Options:
- \(n\)
- \(2n – \frac{1}{2}\)
- \(2n + \frac{1}{2}\)
- \(n + 1\)
- \(n\)
- \(2n – \frac{1}{2}\)
- \(2n + \frac{1}{2}\)
- \(n + 1\)
Correct Answer: a)
Explanation:
In the expansion of \((1+x)^{2n+1}\):
Coefficient of \(x^r\) is \({}^{2n+1}C_r\).
Coefficient of \(x^{r+1}\) is \({}^{2n+1}C_{r+1}\).
Given \({}^{2n+1}C_r = {}^{2n+1}C_{r+1}\).
Since \({}^N C_A = {}^N C_B \implies A = B\) or \(A + B = N\):
Here \(r \ne r+1\), so:
\[r + (r+1) = 2n + 1 \implies 2r + 1 = 2n + 1 \implies 2r = 2n \implies r = n\]
Hence, the correct option is (a).
\((1+x)^{2n+1}\) के प्रसार में:
\(x^r\) का गुणांक \({}^{2n+1}C_r\) है तथा \(x^{r+1}\) का गुणांक \({}^{2n+1}C_{r+1}\) है।
प्रश्नानुसार:
\[{}^{2n+1}C_r = {}^{2n+1}C_{r+1}\]
चूँकि \({}^nC_x = {}^nC_y \implies x + y = n\):
\[r + (r+1) = 2n + 1 \implies 2r + 1 = 2n + 1 \implies r = n\]
अतः सही विकल्प (a) है।