In an A.P., the fourth and sixth terms are 8 and 14 respectively. Find the :

(i) first term

(ii) common difference

(iii) sum of first 20 terms

Given, a₄ = 8 and a₆ = 14.

By formula, a_n = a + (n - 1)d
⇒ a₄ = a + (4 - 1)d
⇒ 8 = a + 3d
⇒ a = 8 - 3d     (Eq 1)

⇒ a₆ = a + (6 - 1)d
⇒ 14 = a + 5d
Putting value of a from Eq 1 in above equation
⇒ 14 = 8 - 3d + 5d
⇒ 14 = 8 + 2d
⇒ 2d = 14 - 8
⇒ 2d = 6
⇒ d = 3.

Putting value of d in Eq 1,
⇒ a = 8 - 3(3)
⇒ a = 8 - 9
⇒ a = -1.

(i) Hence, the first term of the A.P. is -1.

(ii) Hence, the common difference of the A.P. = 3.

(iii) By formula S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

⇒ S₂₀ = $\dfrac{20}{2}[2 \times (-1) + (20 - 1)3]$
⇒ S₂₀ = 10[-2 + 57]
⇒ S₂₀ = 10 × 55
⇒ S₂₀ = 550.

Hence, the sum of first 20 terms of the A.P. is 550.