Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Sum of the Coefficients of Two Middle Terms in the Binomial Expansion of ( 1 + X ) 2 N − 1 - Mathematics

Find the sum of the coefficients of two middle terms in the binomial expansion of  $\left( 1 + x \right)^{2n - 1}$

#### Solution

$\left( 1 + x \right)^{2n - 1}$
$\text{ Here, n is an odd number .}$
$\text{ Therefore, the middle terms are } \left( \frac{2n - 1 + 1}{2} \right)^{th} \text{ and } \left( \frac{2n - 1 + 1}{2} + 1 \right)^{th} , i . e . , n^{th} \text{ and } (n + 1 )^{th} \text{ terms } .$
$\text{ Now, we have}$
$T_n = T_{n - 1 + 1}$
$=^{2n - 1}{}{C}_{n - 1} \left( x \right)^{n - 1}$
$\text{ And } ,$
$T_{n + 1} = T_{n + 1}$
$= ^{2n - 1}{}{C}_n \left( x \right)^n$
$\therefore \text{ the coefficients of two middle terms are } ^{2n - 1}{}{C}_{n - 1} \text{ and } ^{2n - 1}{}{C}_n .$
$Now,$
$^{2n - 1} C_{n - 1} +^{2n - 1} C_n =^{2n} C_n$

Hence, the sum of the coefficients of two middle terms in the binomial expansion of

$\left( 1 + x \right)^{2n - 1}$ is ${}^{2n} C_n$ .

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Q 12 | Page 45