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

Question

Between 1 and 31 are inserted m arithmetic means, so that the ratio of the 7 text th and (m-1) th means is 5: 9. Then find the value of m. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

d=(31-1/m+1)=(30/m+1) (A7/Am-1)=(1+7 d/31-2 d)=(5/9) ⇒ d=2=(30/m+1) ⇒ m=14

Q. Between $1$ and $31$ are inserted $m$ arithmetic means, so that the ratio of the $7^{th}$ and $(m - 1)$ th means is $5 : 9$ . Then find the value of $m$ .

$d = \frac{31 - 1}{m + 1} = \frac{30}{m + 1}$
$\frac{A _{7}}{A _{m - 1}} = \frac{1 + 7 d}{31 - 2 d} = \frac{5}{9}$
$\Rightarrow d = 2 = \frac{30}{m + 1}$
$\Rightarrow m = 14$

Q. 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 find the value of m.

d=m+131−1​=m+130​ Am−1​A7​​=31−2d1+7d​=95​ ⇒d=2=m+130​ ⇒m=14

Q. Between $1$ and $31$ are inserted $m$ arithmetic means, so that the ratio of the $7^{th}$ and $(m - 1)$ th means is $5 : 9$ . Then find the value of $m$ .

$d = \frac{31 - 1}{m + 1} = \frac{30}{m + 1}$
$\frac{A _{7}}{A _{m - 1}} = \frac{1 + 7 d}{31 - 2 d} = \frac{5}{9}$
$\Rightarrow d = 2 = \frac{30}{m + 1}$
$\Rightarrow m = 14$