Determine the A.P whose third term is 16 and the difference of 5 ^thfrom7 ^thterm is 12.

Given: T ₃ = 16

T ₇ – T ₅ = 12
Let the A.P. be a, a + d, a + 2d, ……..
⇒ T ₃ = a + 2d = 16 ………………(i)
and

⇒ T ₇ = a + 6d

and

⇒ T ₅ = a + 4d

∵ T ₇ – T ₅ = 12
⇒ (a + 6d) – (a + 4d) = 12
⇒ a + 6d – a – 4d = 12
⇒ 2d = 12
⇒ d = 6

Substituting the value of d = 6 in (i),
a + 12 = 16

⇒ a = 4

∴ The first term of the A.P. is 4 and the common difference is 6.
∴ The A.P. is 4, 10, 16, 22, 28, 34, …