Question

Given a_n = 4, d = 2, S_n = -14, find n and a.

Answer 1

Given a_n = a + (n – 1) d = 4 ……. (1) 

d = 2; S_n = – 14 

From (1); a + (n – 1) 2 = 4 

a = 4 – 2n + 2 

a = 6 – 2n 

Given a = 2, d = 8, S_n = 90 

S_n = \(\frac{n}{2}\)[a + a_n ] 

-14 = \(\frac{n}{2}\)[(6-2n) + 4] [∵ a = 6 – 2n] 

-14 × 2 = n (10 – 2n)

⇒ 10n – 2n² = – 28

⇒ 2n² – 10n – 28 = 0 

⇒ n² – 5n – 14 = 0 

⇒ n² – 7n + 2n – 14 = 0 

⇒ n (n – 7) + 2 (n – 7) = 0 

⇒ (n – 7) (n + 2) = 0 

⇒ n = 7 (or) n = – 2 ∴ n = 7 

Now a = 6 – 2n = 6 – 2 × 7 

= 6 – 14 = -8 

∴ a = – 8; n = 7