Question

In the expansion of $(1+x{)}^{23}$, if $r^{th}$ ,$(r+1{)}^{th}$ , and $(r+2{)}^{th}$ terms are in A.P., then value of ${}^{23}{C}_r$ equals

• A. ${}^{23}{C}_8$
• B. ${}^{23}{C}_7$
• C. ${}^{23}{C}_9$
• D. None of these

Answer 1

Answer: (C)

${}^{23}{C}_9$

Hence

${}^n{C}_{r-1}$ ${}^n{C}_r$ and ${}^n{C}_{r+1}$ are in A.P.

For the above terms to be, in A.P, they must follow

$(n-2r{)}^2=n+2$

Replacing n by 23, we get

$529+4r^2-92r=23+2$

$4r^2-92r+504=0$

$r^2-23r+126=0$

$(r-14)(r-9)=0$

$r=14$ and $r=9$

${}^{23}{C}_r$$={}^{23}{C}_9$