QJEEAdv16Ma8ecd: JEE Advanced 2016 Mathematics - Class 11 JEEAdv16Ma8ecd Solution: Let be the smallest positive integer such that the coefficient…

Question

Verified Solution • 2025–26 Session moderate

Let

m

be the smallest positive integer such that the coefficient of

{x^2}

in the expansion of

{\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,

is

\left( {3n + 1} \right)\,{}^{51}{C_3}

for some positive integer

n

. Then the value of

n

is

Accepted Answer

It is given that the coefficient of

x^2

in

(1+x)^2+(1+x)^3+

\cdots+(1+x)^{49}+(1+m x)^{50}

is

(3 n+1){ }^{51} C_3

.

Now,

\begin{aligned}&amp; { }^2 C_2+{ }^3 C_2+{ }^4 C_2+\cdots+{ }^{49} C_2+m^2{ }^{50} C_2=(3 n+1){ }^{51} C_3 \\\\&amp; \Rightarrow{ }^3 C_3+{ }^3 C_2+{ }^4 C_2+\cdots+{ }^{49} C_2+m^2{ }^{50} C_2=(3 n+1){ }^{51} C_3 \\\\&amp; \quad \quad\left(\text { as } n C r+n C r-1=n^{+1} C r\right) \\\\&amp; \Rightarrow{ }^4 C_3+{ }^4 C_2+\cdots+{ }^{49} C_2+m^{250} C_2=(3 n+1){ }^{51} C_3 \\\\&amp; \Rightarrow{ }^{49} C_3+{ }^{49} C_2+m{ }^{250} C_2=(3 n+1){ }^{51} C_3 \\\\&amp; \Rightarrow{ }^{50} C_3+m^2{ }^{50} C_2=(3 n+1){ }^{51} C_3 \\\\&amp; \Rightarrow{ }^{50} C_3+{ }^{50} C_2+m^2{ }^{50} C_2=(3 n+1){ }^{51} C_3+{ }^{50} C_2 \\\\&amp; \Rightarrow{ }^{51} C_3+m^2{ }^{50} C_2=3 n{ }^{51} C_3+{ }^{51} C_3+{ }^{50} C_2 \\\\&amp; \Rightarrow{ }^{-50} C_2+m^{250} C_2=3 n{ }^{51} C_3 \\\\&amp; \Rightarrow{ }^{50} C_2\left(m^2-1\right)=3 n \cdot \frac{51}{3}{ }^{50} C_2\left({ }^n C_r=\frac{n}{r} \quad{ }^{n-1} C_{r-1}\right) \\\\&amp; \Rightarrow 2\left(m^2-1\right)=51\end{aligned}

Therefore,

m^2=51 n+1

.

\begin{gathered}\Rightarrow n=\frac{m^2-1}{51} \\\\\Rightarrow m=16 \Rightarrow n=5\left(m, n \in \mathrm{I}^{+}\right)\end{gathered}

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