In an A.P., given a = 8, a_n = 62, S_n = 210, find n and d.

Given,

a_n = 62

By formula,

a_n = a + (n - 1)d

Substituting values we get :

⇒ 62 = 8 + (n - 1)d

⇒ (n - 1)d = 54 ……..(1)

Given,

S_n = 210

By formula,

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

Substituting values we get :

$\Rightarrow \dfrac{n}{2}[2 \times 8 + (n - 1)d] = 210 \\[1em] \Rightarrow \dfrac{n}{2}[16 + 54] = 210 \\[1em] \Rightarrow \dfrac{n}{2} \times 70 = 210 \\[1em] \Rightarrow 35n = 210 \\[1em] \Rightarrow n = \dfrac{210}{35} = 6.$

Substituting value of n in equation (1), we get :

⇒ (6 - 1)d = 54

⇒ 5d = 54

⇒ d = $\dfrac{54}{5}$.

Hence, n = 6 and d = $\dfrac{54}{5}$.