Find the sum of all even numbers between 10 and 100.

Even numbers between 10 and 100 are :

12, 14, 16, ….., 98

Let total number of terms in A.P. be n.

Given,

a = 12

d = 14 - 12 = 2

We know that,

∴ a_n = a + (n - 1)d

⇒ 98 = 12 + (n - 1)2

⇒ 98 - 12 = (n - 1)2

⇒ 86 = (n - 1)2

⇒ $\dfrac{86}{2}$ = (n - 1)

⇒ n - 1 = 43

⇒ n = 43 + 1

⇒ n = 44

We know that,

Sum of n terms of an A.P. is given by,

∴ S_n = $\dfrac{n}{2}$ (a + l)

⇒ S₄₄ = $\dfrac{44}{2}$ (12 + 98)

= 22 (110)

= 2420.

Hence, S₄₄ = 2420.