First slide

Means - A.M/G.M/H.M

Question

Between 1 and 31 are inserted M arithmetic means so that the ratio of the 7th and (M - 1) th means is 5:9, then the value of m is

Moderate

A

13
B

14
C

11
D

12

Solution

let A1, A2,.,. , Am be ,m arithmetic means between 1 and 31.
$\Rightarrow$ Then $A_{1}, A_{2}, \dots, A_{m} 31$ is an AP with common difference.
$d = \frac{31 - 1}{m + 1} = \frac{30}{m + 1}$ $[∵ d = \frac{b - a}{n + 1}]$
$∴ A_{7} = 1 + 7 d$
$= 1 + \frac{7 \times 30}{m + 1} = \frac{m + 211}{m + 1}$
and $\begin{aligned} A_{m - 1} = 1 + (m - 1) d \\ = 1 + \frac{30}{m + 1} (m - 1) \\ = \frac{31 m - 29}{m + 1} \end{aligned}$
Now, $\begin{array}{l} \frac{A_{7}}{A_{m - 1}} = \frac{5}{9} \\ \frac{m + 211}{31 m - 29} = \frac{5}{9} \\ \Rightarrow 9 m + 1899 = 155 m - 145 \\ \Rightarrow 146 m = 2044 \\ \Rightarrow m = 14 \end{array}$ (given)

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