The third term of an arithmetic progression is 16 and the 7^th term is 12 more than the 5^th term. Find the arithmetic progression.

Let first term be a and common difference be d.
We know that, a_n = a + (n - 1)d

Where,

a = First term of AP
d = Common difference of AP
and no of terms is ‘n’

∴ a₃ = a + (3 - 1)d
⇒ 16 = a + 2d …(i)
and a₇ = a + (7 - 1)d
⇒ a₇ = a + 6d …(ii)
and a₅ = a + (5 - 1)d
⇒ a₅ = a + 4d …(iii)

Given, a₇ = a₅ + 12
⇒ a + 6d = a + 4d + 12 (From (ii) and (iii))

⇒ 6d – 4d = 12
⇒ 2d = 12
⇒ d = 6
Substituting the value of d in eq. (i), we get,
16 = a + 2(6)
⇒ a = 16 – 12 = 4

Hence first term a = 4 and common difference d = 6
∴ a₁ = 4
⇒ a₂ = a₁ + d = 4 + 6 = 10
⇒ a₃ = a₂ + d = 10 + 6 = 16
⇒ a₄ = a₃ + d = 16 + 6 = 22

Hence, The arithmetic progression is 4, 10, 16, 2.