Question

If $a,b,c,d$ are in HP, then $ab+bc+cd$ is equal to

• A. $3ad$
• B. $(a+b)(c+d)$
• C. $3ac$
• D. none of these

Answer 1

The correct option is A

$3ad$

Find $ab+bc+cd$

Given that $a,b,c,d$ are in Harmonic Progression.

So,

$\begin{array}{ll}b& =\frac{2ac}{a+c}\\ \Rightarrow a+c& =\frac{2ac}{b}\\ \mathrm{And}c& =\frac{2bd}{b+d}\\ \Rightarrow b+d& =\frac{2bd}{c}\\ \Rightarrow (a+c)(b+d)& =\frac{2ac}{b}\frac{2bd}{c}\\ \Rightarrow ab+bc+ad+cd& =4ad\\ \Rightarrow ab+bc+cd& =3ad\end{array}$

Thus, $ab+bc+cd=3ad$.

Hence, option $\left(A\right)$ is the correct option.