Question

The value of $C_0^2+3C_1^2+5C_2^2+\cdots$ up to $51$ terms is equal to
( where $C_r={}^{50}{C}_r$)

• A. $51\cdot 2^{50}$
• B. $102\cdot {}^{100}{C}_{50}$
• C. $51\cdot {}^{100}{C}_{50}$
• D. $99\cdot {}^{100}{C}_{50}$

Answer 1

The correct option is C $51\cdot {}^{100}{C}_{50}$

Let $S=C_0^2+3C_1^2+5C_2^2+\dots +101C_{50}^2$
Now,
$S=101C_{50}^2+99C_{49}^2+97C_{48}^2+\dots +C_0^2$
Adding both, we get
$\Rightarrow 2S=102[C_0^2+C_1^2+C_2^2+\dots +C_{50}^2]\Rightarrow S=51{}^{100}{C}_{50}$