Question

Find the sum of odd natural numbers from 1 to 101

Answer 1

The odd natural numbers from 1 to 101 are

1, 3, 5, ....., 101

The above sequence is an A.P.

∴ a = 1, d = 3 – 1 = 2

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

Then, t_n = 101

Since t_n = a + (n – 1)d,

101 = 1 + (n – 1)(2)

∴ 101 = 1 + 2n – 2

∴ 101 = 2n – 1

∴ 102 = 2n

∴ n = `102/2` = 51

Now, S_n = `"n"/2 ("t"_1 + "t"_"n")`

∴ S₅₁ = `51/2 (1 + 101)`

= `51/2 (102)`

= 51 × 51

= 2601

∴ The sum of odd natural numbers from 1 to 101 is 2601.