Question

If $a,a_1,a_2,,...,a_{2n},b$are in arithmetic progression and $a,g_1,g_2,..,g_{2n},b$are in geometric progression and is the harmonic mean of $a$ and $b$ is

• A. $2nh$
• B. $n/h$
• C. $nh$
• D. $2n/h$

Answer 1

The correct option is D

$2n/h$

Explanation for the correct option:

Finding the value:

Given $a,a_1,a_{2}1,\dots ,a_{2n},b$are in arithmetic progression so $a_1,a_{2}1,\dots ,a_{2n}$ are AMs between $a,b$

so, $a_1+a_{2n_{}}=a_2+a_{2n-1}=a+b$

Given $a,g_1,g_2,..,g_{2n},b$are in GP.

so $g_1{g}_{2n}=g_2{g}_{2n-1}=g_n{g}_{n+1}=ab$

$$\begin{gathered} \frac{(a_1+a_{2n})}{\left(g_1{g}_{2n}\right)}+\frac{(a_2+a_{2n-1})}{\left(g_2{g}_{2n-1}\right)}+.............+\frac{(a_n+a_{n+1})}{\left(g_n{g}_{n+1}\right)}=\frac{a+b}{ab}+\frac{a+b}{ab}+.....\frac{a+b}{ab} \\ =n\left(\frac{a+b}{ab}\right) \\ =\raisebox{1ex}{$2n$}\!\left/ \!\raisebox{-1ex}{$hwhereh=\left({\displaystyle \frac{2ab}{a+b}}\right)$}\right. \end{gathered}$$

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