Mathematics - Binomial Theorem Question with Solution | TestHub

Coefficient of$x^2$in the expansion of
${\left(1+x\right)}^2+{\left(1+x\right)}^3+\dots {\left(1+x\right)}^{49}+{\left(1+mx\right)}^{50}$is
${}^2{C}_2+{}^3{C}_2+\dots {}^{49}{C}_2+{}^{50}{C}_2{m}^2=\left(3n+1\right){}^{51}{C}_3$
⇒${}^3{C}_3+{}^3{C}_2+\dots {}^{49}{C}_2+{}^{50}{C}_2{m}^2=\left(3n+1\right){}^{51}{C}_3$(Use${}^n{C}_r{+}^n{C}_{r+1}{=}^{n+1}{C}_{r+1}$)
⇒${}^{50}{C}_3+{}^{50}{C}_2{m}^2=\left(3n+1\right) {}^{51}{C}_3$
$\frac{\mathrm{50.49.48}}{6}+\frac{50.49}{2} m^2=\left(3n+1\right)\frac{\mathrm{51.50.49}}{6}$
$m^2=51n+1$must be a perfect square
$By\; trial\; \Rightarrow$$n=5$and$m=16$($\mathrm{M},\mathrm{n}\in \mathrm{N}$)
$\Rightarrow n=5$