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

Given,

a₅ + a₇ = 52   (Eq 1)

a₁₀ = 46         (Eq 2)

By using formula a_n = a + (n - 1)d for Eq 1 we get,
⇒ a₅ + a₇ = 52
⇒ a + (5 - 1)d + a + (7 - 1)d = 52
⇒ a + 4d + a + 6d = 52
⇒ 2a + 10d = 52
⇒ 2(a + 5d) = 52
⇒ a + 5d = 26
⇒ a = 26 - 5d    (Eq 3)

By using formula a_n = a + (n - 1)d for Eq 2 we get,
⇒ a₁₀ = 46
⇒ a + (10 - 1)d = 46
⇒ a + 9d = 46

Putting value of a from Eq 3 in above equation
⇒ 26 - 5d + 9d = 46
⇒ 26 + 4d = 46
⇒ 4d = 46 - 26
⇒ 4d = 20
⇒ d = 5.

∴ a = 26 - 5d = 26 - 5 × 5 = 26 - 25 = 1.

We know a₁ = a so,

a₂ = a₁ + d = 1 + 5 = 6,
a₃ = a₂ + d = 6 + 5 = 11,
a₄ = a₃ + d = 11 + 5 = 16.

Hence, the required A.P. is 1, 6, 11, 16, 21, …