Question

In the binomial expansion of $ (p - q)^{14} $, if the sum of 7^th and 8^th terms is zero, then $$ \frac{p + q}{p - q} = ? $$

• A. 14
• B. 15
• C. 16
• D. 13

Hint:

Use the general term formula for binomial expansions and simplify term ratios when given relationships.

Answer 1

The Correct Option is B

The general term in binomial expansion is: \[ T_{r+1} = \binom{14}{r} p^{14 - r} (-q)^r \] 7^th term: \( T_7 = \binom{14}{6} p^8 (-q)^6 = \binom{14}{6} p^8 q^6 \)
8^th term: \( T_8 = \binom{14}{7} p^7 (-q)^7 = -\binom{14}{7} p^7 q^7 \)
Sum is zero: \[ \binom{14}{6} p^8 q^6 - \binom{14}{7} p^7 q^7 = 0 \Rightarrow \frac{\binom{14}{6}}{\binom{14}{7}} = \frac{p^7 q^7}{p^8 q^6} \Rightarrow \frac{7}{8} = \frac{q}{p} \Rightarrow \frac{p}{q} = \frac{8}{7} \Rightarrow \frac{p + q}{p - q} = \frac{8 + 7}{8 - 7} = \frac{15}{1} = 15 \]