Question

If the (p+1)th, (q+1)th, (r+1)th terms of an AP are in GP when p, q, r are in HP, then the ratio of the first term to the common difference of the AP is

• A. 2q
• B. -2q
• C. -q/2
• D. q/2

Answer 1

The correct option is C

-q/2

Let the terms be a+pd, a+qd, a+rd where a is the first term of AP and d is common difference of AP. Now p,q,r are in HP, So, $q=\frac{2pr}{(p+r)}$ and also the terms are in GP. So $(a+qd{)}^2=(a+pd)\times (a+rd)$. Solving this leaves us with $\left(\frac{a}{d}\right)=\frac{(q^2-rp)}{(p+r-2q)}$. Substituting $p+r=\frac{2pr}{q}$, we get $\frac{a}{d}=\frac{-q}{2}$ .Hence, choice (c) is the right option