Q35 / 90 · JEE Main 2022 (28 Jun Shift 2) · Mathematics

If $n$ arithmetic means are inserted between a and $100$ such that the ratio of the first mean to the last mean is $1 : 7$ and $a + n = 33$ , then the value of $n$ is

Solution

Let the A.P. be $a, A_{1}, A_{2} . . . A_{n}, 100$

Here, common difference, $d = \frac{100 - a}{n + 1}$

Given $\frac{A_{1}}{A_{n}} = \frac{1}{7} \Rightarrow \frac{a + d}{100 - d} = \frac{1}{7} . . . (i)$

Also $a + n = 33$

From options, when $n = 23$ , $a = 10 and d = \frac{90}{24} = \frac{15}{4}$

from $(i)$ $\frac{10 + \frac{15}{4}}{100 - \frac{15}{4}} = \frac{55}{385} = \frac{1}{7}$

Hence, the correct answer is option: 'C'.