Question

Which number should be added to the numbers $13,15,19$ so that the resulting numbers be the consecutive terms of an H.P.?

• A. $7$
• B. $6$
• C. $-6$
• D. $-7$

Answer 1

The correct option is D

$-7$

Explanation for the correct options:

Finding the missing number

Let us consider $x$ be the number added to $13,15,$ and $19$.

Therefore, $13+x,15+x$ and $19+x$ are in HP.

Then, $\frac{1}{13+x},\frac{1}{15+x},\frac{1}{19+x}$ are in AP.

$\frac{2}{15+x}=\frac{1}{13+x}+\frac{1}{19+x}$

$\Rightarrow \frac{2}{15+x}=\frac{19+x+13+x}{(13+x)(19+x)}$

$\Rightarrow \frac{2}{15+x}=\frac{32+2x}{(13+x)(19+x)}$

$\Rightarrow (15+x)(2x+32)=2(13+x)(19+x)$

$\Rightarrow 30x+2x^2+480+32x=2(247+13x+19x+x^2)$

$\Rightarrow 2x^2+62x+480=2x^2+64x+494$

$\Rightarrow 2x=-14$

$\Rightarrow x=\frac{-14}{2}$

$\Rightarrow x=-7$

Therefore, option (d) is correct option.