Given:
a4 = 3a1 …(i)
a7 = 2a3 +1 …(ii)
We know
an = a + (n -1) d
So,
a3 = a +2d
a4 = a +3d
a7 = a +6d
Put all the values of a1 and a4 in (i),
a + 3d = 3a
⇒ 3d = 3a - a
⇒ 3d = 2a
⇒ d = \(\frac{2a}{3}\)
Put all the values of a7 and a3 in (ii),
⇒ a + 6d = 2(a +2d) +1
a + 6d = 2a + 4d + 1
⇒ 6d - 4d = 2a - a +1
⇒ 2d = a +1
⇒ d = 2
Hence, first term is 3 and the common difference is 2.