Find the sum of all odd numbers between 100 and 150.

The odd numbers between 100 and 150 are :

101, 103, 105,…….., 149.

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

We know that,

∴ a_n = a + (n - 1)d

Given,

a = 101

a_n = 149

d = 103 - 101 = 2

⇒ 149 = 101 + (n - 1)2

⇒ 149 - 101 = (n - 1)2

⇒ 48 = (n - 1)2

⇒ $\dfrac{48}{2}$ = (n - 1)

⇒ 24 = (n - 1)

⇒ n = 24 + 1

⇒ n = 25.

We know that,

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

∴ S_n = $\dfrac{n}{2}$ (a + l)

⇒ S₂₅ = $\dfrac{25}{2}$ (101 + 149)

= $\dfrac{25}{2} \times (250)$

= 25 × 125

= 3125.

Hence, S₂₅ = 3125.