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

Given: a₈ = 0

To prove: a₃₈ = 3 × a₁₈

Now, Consider a₈ = 0

⇒ a + (8 - 1)d = 0

⇒ a + 7d = 0

⇒ a = - 7d ……. (1)

Consider a₃₈ = a + (38 - 1)d

⇒ a₃₈ = - 7d + 37d (from equation (1))

⇒ a₃₈ = 30d ………. (2)

Now, consider a₁₈ = a + (18 - 1)d

⇒ a₁₈ = - 7d + 17d (from equation (1))

⇒ a₁₈ = 10d ……….(3)

From equation (2) and (3), we get,

a₃₈ = 3 × a₁₈

Hence, proved.