Between $1$ and $31, m$ numbers have been inserted in such a way that the resulting sequence is an $A. P.$ and the ratio of $7^{\text {th }}$ and $(m-1)^{\text {th }}$ numbers is $5: 9 .$ Find the value of $m$
Between $1$ and $31, m$ numbers have been inserted in such a way that the resulting sequence is an $A. P.$ and the ratio of $7^{\text {th }}$ and $(m-1)^{\text {th }}$ numbers is $5: 9 .$ Find the value of $m$
Let $A_{1}, A_{2}, \ldots \ldots A_{m}$ be m numbers such that $1, A_{1}, A_{2}, \ldots \ldots A_{m}, 31$ is an $A.P.$
Here, $a=1, b=31, n=m+2$
$\therefore 31=1+(m+2-1)(d)$
$\Rightarrow 30=(m+1) d$
$\Rightarrow d=\frac{30}{m+1}$ ...........$(1)$
$A_{1}=a+d$
$A_{2}=a+2 d$
$A_{3}=a+3 d$
$\therefore A_{7}=a+7 d$
$A_{m-1}=a+(m-1) d$
According to the given condition,
$\frac{a+7 d}{a+(m-1) d}=\frac{5}{9}$
$\Rightarrow \frac{1+7\left(\frac{30}{(m+1)}\right)}{1+(m-1)\left(\frac{30}{m+1}\right)}=\frac{5}{9}$ [ From $(1)$ ]
$\Rightarrow \frac{m+1+7(30)}{m+1+30(m-1)}=\frac{5}{9}$
$\Rightarrow \frac{m+1+210}{m+1+30 m-30}=\frac{5}{9}$
$\Rightarrow \frac{m+211}{31 m-29}=\frac{5}{9}$
$\Rightarrow 9 m+1899=155 m-145$
$\Rightarrow 155 m-9 m=1899+145$
$\Rightarrow 146 m=2044$
$\Rightarrow m=14$
Thus, the value of $m$ is $14$
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