How many terms are there in an AP whose first and fifth

How many terms are there in an AP whose first and fifth terms are –14 and 2 respectively and the sum of terms is 40 ?

Here , first term = –14 and fifth term = 2
According to question,
T_n = a + (n – 1).d
T₅ = a + (5 – 1).d
2 = – 14 + 4d

d =	16	= 4
	4

∴ S_n =	n	[2a + (n - 1) × d]
	2

40 =	n	[- 28 + (n - 1) × 4]
	2

⇒ 80 = – 28n + 4n² – 4n
⇒ 4n² – 32n – 80 = 0
n² – 8n – 20 = 0
⇒ (n – 10)(n + 2) = 0
∴ n = 10 (∵ n ≠ -2)