Question

Given the ratio of sum of n terms of two AP's = (7n+1):(4n+27). Find the ratio of their $m^{th}$ terms.

Answer 1

Given ratio of sum of n terms of two AP's = (7n+1):(4n+27)
Let's consider the ratio these two AP's mth terms as $a_m:a_m^{\prime }\to$ (2)
Recall the nth term of AP formula, $a_n=a+(n-1)d$
Hence, equation (2) becomes,
$a_m:a_m^{\prime }=a+(m-1)d:a^{\prime }+(m-1)d^{\prime }$
On multiplying by 2, we get
$a_m:a^{\prime }m=[2a+2(m-d\left)d\right]:[2a^{\prime }+2(m-1\left)d^{\prime }\right]$
$=[2a+\{(2m-1)-1\left\}d\right]:[2a^{\prime }+\{(2m-1)-1\left\}{d}^{\prime }\right]$
$=S_{2m-1}:S_{2m-1}^{\prime }$
$=\left[7\right(2m-1)+1]:\left[4\right(2m-1)+27]$ [from (1)]
$=[14m-7+1]:[8m-4+27]$
$=[14m-6]:[8m+23]$
Thus the ratio of mth terms of two AP's is [14m - 6] : [8m + 23].