Find the sum of all odd integers from 1 to 2001.

The odd numbers lying between 1 and 2001 are

1, 3, 5,…, 2001

a₂ – a₁ = 3 – 1 = 2

a₃ – a₂ = 5 – 3 = 2

∵ a₃ – a₂ = a₂ – a₁ = 2

Therefore, the series is in AP

Here, a = 1, d = 2 and a_n = 2001

We know that,

a_n = a + (n – 1)d

⇒ 2001 = 1 + (n – 1)2

⇒ 2001 – 1 = (n – 1)2

⇒ 2000 = (n – 1)2

⇒ 1000 = (n – 1)

⇒ n = 1001

Now, we have to find the sum of this AP

⇒ S₁₀₀₁ = 1001[1 + 1000]

⇒ S₁₀₀₁ = 1001 [1001]

⇒ S₁₀₀₁ = 1002001

Hence, the sum of all odd numbers lying between 1 and 2001 is 1002001.