The 4^th term of an A.P. is three times the first and the 7^th term exceeds twice the third term by 1. Find the first term and the common difference.

Given:

a₄= 3a₁ …(i)

a₇= 2a₃ +1 …(ii)

a₃ = a +2d

a₄ = a +3d

a₇ = a +6d

Put all the values of a₁ and a₄ in (i),

a + 3d = 3a

3d = 2a

d=

Put all the values of a₇ and a₃ in (ii),

a + 6d= 2(a +2d) +1

2d = a +1

Put d = ,

2() = a +1

–a =1

a= 3

Now, difference, d=

d= 2

Hence, first term is 3 and the common difference is 2.