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

# If A, B, C and D in Any Binomial Expansion Be the 6th, 7th, 8th and 9th Terms Respectively, Then Prove that B 2 − a C C 2 − B D = 4 a 3 C - Mathematics

If abc and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that $\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}$

#### Solution

$\text{ Suppose the binomial expression is } (1 + x )^n .$

$\text{ Then, the 6th, 7th, 8th and 9th terms are } ^{n}{}{C}_5 x^5 , ^{n}{}{C}_6 x^6 , ^{n}{}{C}_7 x^7 \text{ and }^{n}{}{C}_8 x^8 , \text{ respectively } .$

$\text{ Now, we have: }$

$\frac{^{n}{}{C}_{6 x^6}}{^{n}{}{C}_5 x^5} = \frac{b}{a}, \frac{^{n}{}{C}_{8 x^8}}{^{n}{}{C}_7 x^7} = \frac{d}{c} \text{ and } \frac{^{n}{}{C}_{7 x^7}}{^{n}{}{C}_6 x^6} = \frac{c}{b}$

$\Rightarrow \frac{n - 5}{6} = \frac{b}{a} \text{ and } \frac{n - 6}{7} = \frac{c}{b}$

$\Rightarrow \frac{\frac{n - 6}{7}}{\frac{n - 5}{6}} = \frac{\frac{c}{b}}{\frac{b}{a}}$

$\Rightarrow \frac{6n - 36}{7n - 35} = \frac{c}{a}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 30 | Page 40