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

Given: a₁₃ = 4(a₃)

a₅ = 16

To find: Sum of first ten terms.

Now, Consider a₁₃ = 4a₃

⇒ a + 12d = 4[a + 2d]

⇒ a + 12d = 4a + 8d

⇒ 3a = 4d ………. (1)

Consider a₅ = a + (5 - 1)d = 16

⇒ a + 4d = 16

⇒ a + 3a = 16 (from equation (1))

⇒ 4a = 16

⇒ a = 4 ………. (2)

∴ d = 3

Sum of n terms of an arithmetic series is given by:

S_n = [2a + (n - 1)d]

Therefore sum of 10 terms of the arithmetic series is given by:

∴ S₁₀ = [2(4) + (10 - 1)(3)]

= 5 × [8 + 27]

= 5 × 35

= 175