In an AP, the first term is 2, the last term is 29 and the sum of all the terms is 155. Find the common difference.

Here, first term = a = 2

Let the Common difference = d

Last term = l = 29

Sum of all terms = S_n = 155

Let there be n terms in the AP.

Now, Sum of n terms of this arithmetic series is given by:

S_n = [2a + (n - 1)d]

= [a + a + (n - 1)d]

= [a + l]

Therefore sum of n terms of this arithmetic series is given by:

∴ S_n = [2 + 29] = 155

⇒ 31n = 310

⇒ n = 10

∴ there are 10 terms in the AP.

Thus 29 be the 10^th term of the AP.

∴ 29 = a + (10 - 1)d

⇒ 29 = 2 + 9d

⇒ 27 = 9d

⇒ d = 3

∴ common difference = d =3