Question

If n arithmetic means are inserted between 20 and 80 such that the ratio of first mean to the last mean is 1:3, then the value of n is

• A. 10
• B. 9
• C. 11
• D. 12

Answer 1

The correct option is C 11

Let $A_1,A_2,A_3,A_4,,....,A_n$ be n arithmetic means between 20 and 80 and d be the common difference of the AP thus formed $20,A_1,A_2,A_3,A_4,,....,A_n,80\mathrm{If}m\mathrm{AM}\text{'}s\mathrm{are}\mathrm{inserted}\mathrm{between}a\mathrm{and}b,\mathrm{then}\mathrm{Common}\mathrm{Difference}d=\frac{b-a}{m+1}d=\frac{80-20}{n+1}=\frac{60}{n+1}{A}_1=20+d=20+\frac{60}{n+1}=\frac{20n+80}{n+1}{A}_n=20+\mathrm{nd}=20+n\frac{60}{n+1}=\frac{80n+20}{n+1}\frac{A_1}{A_n}=\frac{1}{3}\Rightarrow \frac{\frac{20n+80}{n+1}}{\frac{80n+20}{n+1}}=\frac{1}{3}\Rightarrow \frac{20(n+4)}{20(4n+1)}=\frac{1}{3}\Rightarrow 3n+12=4n+1\Rightarrow n=11$