Find the A.P. whose 10 ^th term is 5 and 18 ^th termis 77.

Given: 10 ^th term of A.P = T ₁₀ = 5
18 ^th term of A.P = T ₁₈ = 77

Now we know that,

T _n = a + (n – 1)d

Therefore,

5 = a + (10 – 1) d
⇒ a + 9d = 5 … (1)
And,

77 = a + (18 – 1) d
⇒ a + 17d = 77 ….(2)

Now,

Subtract (1) from (2),

⇒ (a + 17d) – (a + 9d) = 77 – 5
⇒ a + 17d – a – 9d = 77 – 5

⇒ 8d = 72

⇒ d = 9

Now,

a + 9d = 5
⇒ a + 9 × 9 = 5
⇒ a + 81 = 5

⇒ a = 5 – 81

⇒ a = – 76

∴

T ₁ = –76 + (1 – 1)9 = –76

T ₂ = –76 + (2 – 1)9 = –67

T ₃ = –76 + (3 – 1)9 = –58

∴ The A.P. is – 76, - 67, –58,………….