Class 11

Math

Algebra

Binomial Theorem

If $(2+3x )_{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are :

- $7_{th}$ and $8_{th}$
- $8_{th}$ and $9_{th}$
- $28_{th}$ and $29_{th}$
- $27_{th}$ and $28_{th}$

Coefficient of $x_{r}$ and $x_{r+1}$ are equal.

Coefficient of $T_{r+1}=$ Coefficient
of $T_{r+2}$

$⇒_{55}C_{r}(2)_{55−r}(1 3)_{r}=_{55}C_{r+1}(2)_{54−r}(1 3)_{r+1}$

.

$55! (55−r)!r!(2)=55! (54−r)!(r+1)!1 3$

$⇒6(r+1)=55−r$

$⇒r=7$

So, the terms will be $8_{th}$ and $9_{th}$