Question

The9th term of an AP is 7 times the second term and 12 term exceeds 5 times the 3rd term by 2. Find the first term and common difference.

Answer 1

let first term of A.P = a

let common difference = d

nth term is given by an = ( a + ( n -1 ) d )

so 9th term = a9 = ( a + ( 9-1 ) d ) = a + 8d

2nd term = a2 = ( a + ( 2-1 ) d ) =  a + d

given that 7 × a2 = a9

⇒ 7 (  a + d ) = a + 8d

⇒ 7 a + 7d  = a + 8d

⇒ 6a - d =  0                       ----------------------------- eq 1

12 term = a12 = a + 11d and 3rd term a3 = a + 2d

given that  a12 - ( 5 × a3 ) = 2

⇒ a + 11d -  5 ( a + 2d )  = 2

⇒ a + 11d - 5a  -10d = 2

⇒ d - 4a   = 2

⇒  d  = 2 + 4a ----------------------------- eq 2

substitutuing value of d from eq 2 in eq 1 we get

⇒6a - ( 4a +2) = 0

⇒6a  - 4a - 2 = 0

⇒ 2a = 2

⇒ a = 1

from eq 2

d = 2 + 4a = 2 + 4 * 1 = 6

so first term = 1 and common difference d = 6