In an AP, the first term is – 4, the last term is 29 and the sum of all its terms is 150. Find its common difference.

Here, first term = a = - 4

Let the Common difference = d

Last term = l = 29

Sum of all terms = S_n = 150

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 = [ - 4 + 29] = 150

⇒ 25n = 300

⇒ n = 12

∴ there are 12 terms in the AP.

Thus 29 is the 12^th term of the AP.

∴ 29 = a + (12 - 1)d

⇒ 29 = - 4 + 11d

⇒ 29 + 4 = 11d

⇒ 11d = 33

⇒ d = 3

∴ Common difference = d =3