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

# If the Coefficients of ( 2 R + 4 ) Th and ( R − 2 ) Th Terms in the Expansion of ( 1 + X ) 18 Are Equal, Find R. - Mathematics

If the coefficients of $\left( 2r + 4 \right)\text{ th and } \left( r - 2 \right)$ th terms in the expansion of  $\left( 1 + x \right)^{18}$  are equal, find r.

#### Solution

$Given:$
$(1 + x )^{18}$
$\text{ We know that the coefficient of the rth term in the expansion of } (1 + x )^n \text{ is } ^{n}{}{C}_{r - 1}$
$\text{ Therefore, the coefficients of the (2r + 4)th and (r - 2)th terms in the given expansion are } ^{18}{}{C}_{2r + 4 - 1} \text{ and } ^{18}{}{C}_{r - 2 - 1}$
$\text{ For these coefficients to be equal, we must have }$
$^{18}{}{C}_{2r + 3} =^{18}{}{C}_{r - 3}$
$\Rightarrow 2r + 3 = r - 3 or, 2r + 3 + r - 3 = 18 [ \because ^{n}{}{C}_r = ^{n}{}{C}_s \Rightarrow r = s \text{ or } r + s = n]$
$\Rightarrow r = - 6 \text{ or } , r = 6$
$\text{ Neglecting negative value We get}$
$r = 6$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 17 | Page 39