Question

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18^th terms.

Answer 1

Let sum of n terms of two A.P be $S_n$ and $S_n^{{}^{\prime }}$.
Then, $S_n=5n+4$ and $S_n^{{}^{\prime }}=9n+16$ respectively.
Then, if ratio of sum of n terms of 2A.P is given, then the ratio of there nth term is obtained by replacing n by (2n-1).
$\frac{a_n}{a_n^{{}^{\prime }}}=\frac{5(2n-1)+4}{9(2n-1)+16}$
$\therefore$ Ratio of there 18th term is
$\frac{a_{18}}{a_{18}^{{}^{\prime }}}=\frac{5(2\times 18-1)+4}{9(2\times 18-1)+16}=\frac{5\times 35+4}{9\times 35+16}=\frac{179}{321}$