The sum of the 5th and the 7th terms of an AP is 52 and the 10th term is 46. Find the AP.

Let the first term, common difference be a and d respectively.

As we know

a_n = a + (n - 1)d

and Given

a₅ + a₇ = 52

a + 4d + a + 6d = 52

2a + 10d = 52

a + 5d = 26

a = 26 - 5d [ eqn i]

also we have given

a₁₀ = 46

a + 9d = 46

26 - 5d + 9d = 46 [ from eqn i]

4d = 46 - 26

4d = 20

d = 5

using this value in eqn I , we get

a = 26 - 5d

a = 26 - 5(5)

a = 1

as a = 6 and d = 5

So, required AP is a, a + d, a + 2d, a + 3d, …. i.e., 1, 1 + 5, 1 + 2(5), 1 + 3(5), …. i.e., 1, 6, 11, 16, …