Question

If $b^2,a^2,c^2$are in A.P., then $a+b,b+c,c+a,$ will be in

• A. A.P
• B. G.P
• C. H.P
• D. None of these

Answer 1

The correct option is C

H.P

Explanation for the correct option:

Finding the terms $a+b,b+c,c+a,$ will be in:

Given $b^2,a^2,c^2$ are in A.P

$\Rightarrow$ $a^2-b^2=c^2-a^2$

$\Rightarrow$ $(a-b)(a+b)=(c-a)(c+a)$

$\Rightarrow$ $\frac{(a-b)}{(c+a)}=\frac{(c-a)}{(a+b)}$

$\Rightarrow$ $\frac{(a+c)-(b+c)}{(c+a)(b+c)}=\frac{(b+c)-(a+b)}{(a+b)(b+c)}$

$\Rightarrow$ $\frac{1}{(b+c)}–\frac{1}{(c+a)}=\frac{1}{(a+b)}–\frac{1}{(b+c)}$

$\Rightarrow$$\frac{1}{(b+c)}–\frac{1}{(a+b)}=\frac{1}{(c+a)}–\frac{1}{(b+c)}$

$\Rightarrow$$\frac{1}{(a+b)},\frac{1}{(b+c)},\frac{1}{(c+a)}$ are in AP

$\therefore a+b,b+c,c+a$ are in HP

Hence, Option ‘C’ is Correct.