The sums of an terms of two arithmetic progressions are in the ratio (7n – 5) : (5n + 17). Show that their 6th terms are equal.

Given: Ratio of sum of n terms of 2 AP’s
To Prove: 6th terms of both AP’S are equal
Let us consider 2 AP series AP1 and AP2.
Putting n = 1, 2, 3… we get AP1 as 12,19,26… and AP2 as 22,27,32….
So, a1 = 12, d1 = 7 and a2 = 22, d2 = 5
For AP1
S6 = 12 + (6 – 1)7 = 47
For AP2
S6 = 22 + (6 – 1)5 = 47
Therefore their 6th terms are equal.
Hence proved.