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Binomial theorem for positive integral Index

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Question

Let m be the smallest positive integer such that the coefficient of $X^{2}$ in the expansion of $(1 + x)^{2} + (1 + x)^{3} + \dots + (1 + x)^{49} + (1 + mx)^{50} is (3 n + 1)^{51} C_{3}$ for some positive integer n. Then the value of n is _________.

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Solution

Coefficient of x²in expansion
$= 1 +^{3} C_{2} +^{4} C_{2} +^{5} C_{2} + \dots +^{49} C_{2} +^{50} C_{2} \cdot m^{2} [{as}^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}]$
$\begin{array}{l} = (^{3} C_{3} +^{3} C_{2}) +^{4} C_{2} +^{5} C_{2} + \dots +^{49} C_{2} +^{50} C_{2} \cdot m^{2} \\ = (^{4} C_{3} +^{4} C_{2}) + \dots +^{50} C_{2} m^{2} \\ =^{5} C_{3} + \dots +^{49} C_{2} +^{50} C_{2} m^{2} \\ =^{50} C_{3} +^{50} C_{2} m^{2} +^{50} C_{2} -^{50} C_{2} \\ =^{51} C_{3} +^{50} C_{2} (m^{2} - 1) \\ = (3 n + 1) 51_{C_{3}} (given) \end{array}$
$∴ 3 n \cdot {\frac{51}{3}}^{50} C_{2} =^{50} C_{2} (m^{2} - 1)$
$\frac{m^{2} - 1}{51} = n$ $\Rightarrow m^{2} - 1 = 51 n$ $\Rightarrow m^{2} = 51 n + 1$
n=5 $\Rightarrow$ $m^{2} = 256$
value of n is 5.

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