Question

Let $A_1,A_2,...A_{n}and{H}_1,H_2,...H_n$ are $n$ $A.Ms$ & $H.Ms$ between the two quantities $a$ & $b$ $(a,b>0)$ respectively. On the basis is of above information answer the following questionsLet $q$ be the first $H.M$ between $a$& $b,$ then $q$ equals

• A. $\frac{ab(n+1)}{a+bn}$
• B. $\frac{ab(n+1)}{b+an}$
• C. $\frac{ab(n-1)}{a+bn}$
• D. $\frac{ab(n-1)}{b+an}$

Answer 1

The correct option is A $\frac{ab(n+1)}{a+bn}$

As $a,H_1,H_2,...,H_n,b\in H.P$

$\Rightarrow \frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},....,\frac{1}{b}\in A.P$

$\therefore$ $Commondifference\left(D\right)$ $=\frac{a-b}{(n+1)ab}$

$\therefore so\frac{1}{H_1}=t_1+d=\frac{1}{a}+\frac{a-b}{(n+1)ab}=\frac{nb+a}{ab(n+1)}$

$\therefore q=H_1=\frac{(n+1)ab}{nb+a}$........................(i)

Hence choice (A) is correct answer.