If the ratio of sum of n terms of two AP's is (7n+1):(4n+27), find the ratio of their mth terms.

Question : If the ratio of sum of n terms of two AP's is (7n+1):(4n+27), find the ratio of their mth terms. [(14m-6):(8m+23)]

Doubt by Mayank

Solution : 

We know 
S_n=n/2[2a+(n-1)d]

ATQ
S_n/S'_n = n/2[2a+(n-1)d]/n/2[2a'+(n-1)d']
S_n/S'_n = [2a+(n-1)d]/[2a'+(n-1)d']
(7n+1)/(4n+27) = [2a+(n-1)d]/[2a'+(n-1)d']

Putting n = 2m-1
(7[2m-1]+1)/(4[2m-1]+27) = [2a+(2m-1-1)d]/[2a'+(2m-1-1)d']
(14m-7+1)/(8m-4+27) = [2a+(2m-2)d]/[2a'+(2m-2)d']
(14m-6)/(8m+23) = [2a+2(m-1)d]/[2a'+2(m-1)d']
(14m-6)/(8m+23) = 2[a+(m-1)d]/2[a'+(m-1)d']
(14m-6)/(8m+23) = [a+(m-1)d]/[a'+(m-1)d']
 (14m-6)/(8m+23) = a_m/a'_m [∵ an=a+(n-1)d]

∴ a_m:a'_m = (14m-6):(8m+23)