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

Given: S₇ = 182

4th and 17th terms are in the ratio 1 : 5.

i.e. [a + 3d] : [(a + 16d] = 1 : 5

⇒ =

⇒ 5(a + 3 d) = (a + 16d)

⇒ 5a + 15d = a + 16d

⇒ 4a = d

Now, consider S₇ = 182

⇒ (7/2)[2a + (7 - 1)d] = 182

⇒ (7/2)[2a + 6(4a)] = 182

⇒ 7 × [26a] = 182 × 2

⇒ 182a = 364

⇒ a = 2

∴ d = 4a

⇒ d = 8

Thus the AP will be a, a + d, a + 2d,…

i.e. AP is 2, 10, 18, 26,….