Question

If the $(n+1{)}^{th}$ term of a harmonic progression is twice the $(3n+1{)}^{th}$ term. Find the ratio of the first term to the $(n+1{)}^{th}$ term

• A. $1:3$
• B. $1:2$
• C. $3:1$
• D. $4:1$
• E. $2:1$

Answer 1

The correct option is E $2:1$

Let the first term of the HP be $\frac{1}{a}$.

Then the nth term will be $\frac{1}{a+(n-1)d}$

$(n+1{)}^{th}$ term $=\frac{1}{a+\left(n\right)d}$

$(3n+1{)}^{th}$ term $=\frac{1}{a+\left(3n\right)d}$

Given, $\frac{1}{a+\left(n\right)d}=\frac{2}{a+\left(3n\right)d}a+\left(3n\right)d=2a+\left(2n\right)da=nd$

Ratio of first term to $(n+1{)}^{th}$ term is $\frac{1}{a}:\frac{1}{a+\left(n\right)d}=a+\left(n\right)d:a=2a:a=2:1$