Question

If a, b, c, d are in H.P. , then ab +bc+cd =

• A. ad
• B. 2ad
• C. 3ad
• D. 4ad

Answer 1

The correct option is C 3ad

Since a, b, c, d are in H.P. therefore $\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}$ are in AP.
$\therefore \frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}=\frac{1}{d}-\frac{1}{c}=\lambda \left(say\right)\therefore a-b=\lambda ab,b-c=\lambda bcandc-d=\lambda cd$
Adding the above three, we get,
$a-d=\lambda (ab+bc+cd)\cdots \left(i\right)$
Now $\frac{1}{d}=\frac{1}{a}+3\lambda \Rightarrow a-d=3\lambda ad$
Substituting in (i) , we get ab + bc + cd = 3ad.