Question

The ratio between the sum of $n$ terms of two A.P.'s is $7n+1:4n+27$. Find the ratio between their $n^{th}$ terms

• A. $\frac{14n+21}{8n+6}$
• B. $\frac{14n+23}{8n+6}$
• C. $\frac{14n-6}{8n+23}$
• D. $\frac{14n+6}{8n+21}$

Answer 1

The correct option is C $\frac{14n-6}{8n+23}$

From the question, we have

$\frac{S_n}{{S_n}^{\prime }}=\frac{a+\left[(n-1)/2\right]d}{a^{\prime }+\left[(n-1)/2\right]d^{\prime }}=\frac{7n+1}{4n+27}$ .......(1)

$\therefore \frac{T_p}{{T_p}^{\prime }}=\frac{a+(p-1)d}{a^{\prime }+(p-1)d^{\prime }}$

Substitute $\frac{n-1}{2}=p-1$ or $n=2p-1$ in (1)

$\therefore \frac{T_p}{{T_p}^{\prime }}=\frac{7(2p-1)+1}{4(2p-1)+27}=\frac{14p-6}{8p+23}$

Now replacing p by n in the above gives

$\therefore \frac{T_n}{{T_n}^{\prime }}=\frac{14n-6}{8n+23}$

Hence, option C is correct.