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Let \small m be the smallest positive integer such that the coefficients of \small x^2 in the expansion of \small (1+x)^2+(1+x)^3+\ldots+(1+x)^{49}+  \small (1+m x)^{50}$ is $(3 n+1){ }^{51} C_3  for some positive integer \small n. Then the value of \small n, is

Option: 1

5


Option: 2

4


Option: 3

3


Option: 4

2


Answers (1)

best_answer

We find that, 

\begin{aligned} &(1+x)^2+(1+x)^3+\ldots(1+x)^{49}+(1+m x)^{50} \\ &=(1+x)^2\left\{\frac{(1+x)^{48}-1}{(1+x)-1}\right\}+(1+m x)^{50} \\ &= \frac{1}{x}\left\{(1+x)^{50}-(1+x)^2\right\}+(1+m x)^{50} \\ & \therefore \quad \text { Coefficient of } x^2 \text { in } \\ & \quad(1+x)^2+(1+x)^3+\ldots(1+x)^{49}+(1+m x)^{50} \\ &=\text { Coefficient of } x^2 \text { in } \\ &\left.\qquad \frac{1}{x}\left\{(1+x)^{50}-(1+x)^2\right\}+(1+m x)^{50}\right] \\ &=\text { Coefficient of } x^3 \text { in } \\ & \quad\left\{(1+x)^{50}-(1+x)^2\right\}+\text { Coefficient of } x^2 \text { in }(1+m x)^{50} \\ &={ }^{50} C_3+{ }^{50} C_2 m^2 \end{aligned}

It is given that this coefficient is (3 n+1)^{51} C_3 .

\begin{aligned} & \therefore{ }^{50} C_3+{ }^{50} C_2 m^2=(3 n+1)^{51} C_3 \\ & \Rightarrow \quad \frac{50 \times 49 \times 48}{3 \times 2 \times 1}+\frac{50 \times 49}{2 \times 1} m^2=(3 n+1) \frac{51 \times 50 \times 49}{3 \times 2 \times 1} \\ & \Rightarrow 16+m^2=17(3 n+1) \end{aligned}

\Rightarrow m^2=51 n+1

The least value of n for which 51n + 1 is a perfect square is
5 and for this value of n the value of m is 16.
Hence, m = 16 and n = 5.

Posted by

avinash.dongre

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