For an A.P. the sum of its terms is 60, common difference is 2 and last term is 18. Find the number of terms in the A.P.

Given,

Common difference (d) = 2

Last term (l) = 18

Let no. of terms be n.

⇒ l = a + (n - 1)d

⇒ 18 = a + 2(n - 1)

⇒ 18 = a + 2n - 2

⇒ a + 2n = 20

⇒ a = 20 - 2n ………(1)

By formula,

S_n = $\dfrac{n}{2}(a + l)$

Substituting values we get :

$\Rightarrow 60 = \dfrac{n}{2}(a + 18) \\[1em] \Rightarrow 120 = n(a + 18)$

Substituting value of a in above equation, we get :

$\Rightarrow 120 = n(20 - 2n + 18) \\[1em] \Rightarrow 120 = n(38 - 2n) \\[1em] \Rightarrow 120 = 38n - 2n^2 \\[1em] \Rightarrow 2n^2 - 38n + 120 = 0 \\[1em] \Rightarrow 2(n^2 - 19n + 60) = 0 \\[1em] \Rightarrow n^2 - 19n + 60 = 0 \\[1em] \Rightarrow n^2 - 15n - 4n + 60 = 0 \\[1em] \Rightarrow n(n - 15) - 4(n - 15) = 0 \\[1em] \Rightarrow (n - 4)(n - 15) = 0 \\[1em] \Rightarrow n - 4 = 0 \text{ or } n - 15 = 0 \\[1em] \Rightarrow n = 4 \text{ or } n = 15.$

Hence, no. of terms = 4 or 15.