Find the number of terms of the AP 18, , 13, …., and find the sum of all its terms. (CBSE 2013)

Let the first term and common difference of given AP be a and d respectively.

first term(a) = 18 and,

common difference(d) = (n+1)th term - nth term = (31/2)-18

= -(5/2)

last term or nth term(a_n) = -(99/2)

Let the total no. of terms in above A.P be n.

∴ a_n = a+(n-1)×d

⇒ -(99/2) = 18+(n-1)×(-5/2)

⇒ -99 = 36-5(n-1)

⇒ -135 = -5n+5

⇒ 5n = 140

∴ n = 28

Thus, the no. of terms of the given A.P = 28.