Find the sum of all the odd numbers lying between 100 and 200.

The odd numbers lying between 100 and 200 are

101, 103, 105,…, 199

a₂ – a₁ = 103 – 101 = 2

a₃ – a₂ = 105 – 103 = 2

∵ a₃ – a₂ = a₂ – a₁ = 2

Therefore, the series is in AP

Here, a = 101, d = 2 and a_n = 199

We know that,

a_n = a + (n – 1)d

⇒ 199 = 101 + (n – 1)2

⇒ 199 – 101 = (n – 1)2

⇒ 98 = (n – 1)2

⇒ 49 = (n – 1)

⇒ n = 50

Now, we have to find the sum of this AP

⇒ S₅₀ = 25[202 + 49 × 2]

⇒ S₅₀ = 25[300]

⇒ S₅₀ = 7500

Hence, the sum of all odd numbers lying between 100 and 200 is 7500.