Question

If in the expansion of $(1+x{)}^{43}$ the coeff of $(2r+1{)}^{th}$ term is equal to the coeff of $(r+2{)}^{th}$ term, then find the sum of all possible solutions of $r$ :

• A. $11$
• B. $0$
• C. $55$
• D. $15$

Answer 1

The correct option is D $15$

Solution:

We know that in the coefficient of $r^{th}$ term in expansion of $(1+x{)}^n={}^n{C}_{r-1}$

$\therefore$ coefficients of $(2r+1{)}^{th}$ and $(r+2{)}^{th}$ in the expansion $(1+x{)}^{43}$ are ${}^{43}{C}_{2r}$ and ${}^{18}{C}_{r+1}.$

A/q,

${}^{43}{C}_{2r}={}^{43}{C}_{r+1}$

or, $2r=r+1$ or $2r+r+1=43$ $(\because {}^n{C}_r={}^{n}Cs$ then $r=s$ or $r+s=n)$

$2r=r+1$

or, $r=1$ and

$2r+r+1=43$

or, $3r=42$

or, $r=14$

So, sum of all possible solutions of $r=1+14=15$

Hence, D is the correct option.