The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, find the common difference and the number of terms.

Here, first term = a = 5

Let the Common difference = d

Last term = l = 45

Sum of all terms = S_n = 400

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 = [5 + 45] = 400

⇒ 50n = 800

⇒ n = 16

∴ there are 16 terms in the AP.

Thus 45 is the 16^th term of the AP.

∴ 45 = a + (16 - 1)d

⇒ 45 = 5 + 15d

⇒ 40 = 15d

⇒ 15d = 40

⇒ d = 8/3

∴ Common difference = d = 8/3