In an AP, the first term is 22, nth term is – 11 and sum of first n terms is 66. Find n and hence find the common difference.

Here, first term = a = 22

Let the Common difference = d

n^th term = a_n = - 11

Sum of first n terms = S_n = 66

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 + a_n]

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

∴ S_n = [22 + (-11)] = 66

⇒ 11n = 132

⇒ n = 12

∴ there are 12 terms in the AP.

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

∴ - 11 = a + (12 - 1)d

⇒ - 11 = 22 + 11d

⇒ - 11 - 22 = 11d

⇒ 11d = - 33

⇒ d = - 3

∴ Common difference = d = - 3

∴ n = 12, d = - 3