Question

If the ratio of the sum of first n terms of two A.P's is (7n+1): (4n+27), find the of their $m^{th}$ terms

Answer 1

Given ratio of sum of n terms of two AP's =(7n+1):(4n+27)
$\frac{n}{2}(2a+(n-1\left)d\right):\frac{n}{2}(2a^{\prime }+(n-1\left)d^{\prime }\right)=(7n+1):(4n+27)(2a+(n-1\left)d\right):(2a^{\prime }+(n-1\left)d^{\prime }\right)=(7n+1):(4n+27)\to \left(1\right)$
Let's consider the ratio these two AP's mth terms as $a_m:a_m^{\prime }\to \left(2\right)$
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
$a_m:a_m^{\prime }=[2a+2(m-1\left)d\right]:[2a^{\prime }+2(m-1\left)d^{\prime }\right]=[2a+\left\{\right(2m-1)-1\}d]"[2a^{\prime }+\left\{\right(2m-1)-1\}{d}^{\prime }]=S_{2m-1}:S_{2m-1}^{\prime }=\left[7\right(2m-1)+1]:\left[4\right(2m-1)+27]\left[from\right(1\left)\right]=[14m-7+1]:[8m+23]=[14m-6]:[8m+23]$
Thus the ratio of mth terms of two AP's is [14m-6]:[8m+23].