Question

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.

Answer 1

a9=7a2
a + 8d= 7a + 7d (Hence :a2=a+d)
d=6a ...(1)

a12 - 5a3=2
a + 11d-5a-10d=2 (hence:a12=a + 11d, a3=a + 2d)
-4a + d= 2
-4a + 6a=2 (from 1)
2a= 2
a=1


d=6a
d=6*1=6

Answer 2

Let the first term of AP be a and common  difference be d.

a9 = 7a2

⇒ a + 8d = 7(a + d) ...(i)

a12 = 5a3 + 2

⇒ a + 11d = 5(a + 2d) + 2 ...(ii)

From (i),

a + 8d = 7a + 7d

- 6a + d = 0 ...(iii)

From (ii),

a + 11d = 5a + 10d + 2

- 4a + d = 2 ...(iv)

Subtracting (iv) from (iii),

- 2a = - 2

⇒ a = 1

From (iii),

- 6 + d = 0

d = 6