The ninth term of an AP is equal to seven times the second term and twelfth term exceeds five times the third term by 2. Find the first term and the common difference.

Let the first term = a

and common difference = d

We know,

nth term of an AP, a_n = a + (n – 1)d

Given

9^th term of an AP is equal to seven times the second term

⇒ a₉ = 7a₂

⇒ a + (9 – 1)d = 7(a + d)

⇒ a + 8d = 7a + 7d

⇒ d = 6a ……[1]

Also,

Twelfth term exceeds five times the third term by 2

⇒ a₁₂ = 5a₃ + 3

⇒ a + 11d = 5(a + 2d) + 2

⇒ a + 11d = 5a + 10d + 2

⇒ d – 4a = 2

⇒ 6a – 4a = 2 [From 1]

⇒ 2a = 2

⇒ a = 1

From [1],

d = 6a

= 6(1)

= 6

Hence, first term of AP is 1 and common difference is 6.