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 $2n$ arithmetic means be $A_1,A_2,A_3,\dots ,A_{2n}$ between $a$ and $b$.
Then, $A_1+A_2+A_3+\dots +A_{2n}=\frac{a+b}{2}\times 2n$
$=\frac{\frac{13}{6}}{2}\times 2n=\frac{13n}{6}$
Given: $A_1+A_2+A_3+\dots +A_{2n}=2n+1$
$\therefore 2n+1=\frac{13n}{6};$
or $12n+6=13n$
$\therefore n=6$.
$\therefore$ The number of means $=2n=2\times 6=12$.