Question

Between two numbers whose sum is $2 \frac{1}{6}$, an even number of arithmetic means are inserted. If the sum of these means exceeds their number by unity, then the number of means are

• A. 12
• B. 10
• C. 8
• D. None of these

Answer 1

Answer: (A)

Let $2 n$ arithmetic means be $A_{1}, A_{2}, A_{3}, \ldots, A_{2 n}$ between $a$ and $b$.
Then, $A_{1}+A_{2}+A_{3}+\ldots+A_{2 n}=\frac{a+b}{2} \times 2 n$
$=\frac{\frac{13}{6}}{2} \times 2 n=\frac{13 n}{6}$
Given: $A_{1}+A_{2}+A_{3}+\ldots+A_{2 n}=2 n+1$
$\therefore 2 n+1=\frac{13 n}{6} ;$
or $12 n+6=13 n$
$\therefore n=6$.
$\therefore $ The number of means $=2 n=2 \times 6=12$.