Question

Let $m$ arithmetic means are inserted between $1$ and $31$. If the ratio of the $7^{\text{th}}$ mean and $(m-1{)}^{\text{th}}$ mean is $5:9$, then which of the following is/are correct?

• A. The value of $m$ is $14$.
• B. The value of $m$ is $16$.
• C. The sum of all mean's is $256$.
• D. The sum of all mean's is $224$.

Answer 1

Answer: (A), (D)

The correct options are
A The value of $m$ is $14$.
D The sum of all mean's is $224$.

Let $A_1,A_2,\dots ,A_m$ be the $m$ arithmetic means between $1$ and $31$, then
$d=\frac{(b-a)}{(m+1)}\Rightarrow d=\frac{30}{m+1}$

Now,
$\frac{A_7}{A_{m-1}}=\frac{a+7d}{a+(m-1)d}(\because A_n=a+nd)\Rightarrow \frac{5}{9}=\frac{1+\frac{210}{m+1}}{1+\frac{(m-1)\times 30}{m+1}}\Rightarrow \frac{m+211}{31m-29}=\frac{5}{9}\Rightarrow 9m+1899=155m-145\Rightarrow 146m=2044\therefore m=\frac{2044}{146}=14$

Therefore, the sum of all mean's
$=\frac{14}{2}[1+31]=14\times 16=224$