The ratio of their $n$-th term. The sum of $n$ terms of two arithmetic series are in the ratio of $(7n+ 1) : (4n+ 27)$. We have to find the ratio of their $n$-th term.
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I tried to find the ratio by using the formula of summation of A.P.
But it becomes too long due to many variables that is $a_1,a_2,d_1,d_2$

It is actually quite simple. Let $a_1$ and $a_1'$ denote the first terms of the first and second progressions with their common differences $d$ and $d'$ respectively. We thus get $$\frac{S_1}{S_2} = \frac {0.5n (2a_1 +(n-1)d)}{0.5n (2a_1' +(n-1)d')} = \frac {2a_1+(n-1)d}{2a_1' +(n-1)d'} = \frac {7n+1}{4n+27} $$ 
The ratio of the $n$th term of the two AP's can be thus calculated as $$\frac{a_n}{a_n'} = \frac {a_1 +(n-1)d}{a_1'+(n-1)d'} = \frac {2a_1 +((2n-1)-1)d}{2a_1' + ((2n-1)-1)d'} $$ $$=\frac {S_{2n-1}}{S_{2n-1}'} = \frac {14n-6}{8n+23} $$ Hope it helps. 
