Find the sum of all odd natural numbers from 1 to 150.

2. Find the sum of all odd natural numbers from 1 to 150.

Sol. The odd natural numbers from 1 to 150 are: 1, 3, 5, 7, 9, .........., 149.

These numbers form an A.P. with a = 1, d = 2

Let, 149 be nth term of an A.P.

tn = 149

tn = a + (n – 1) d

∴ 149 = 1 + (n – 1) 2

∴ 149 = 1 + 2n – 2

∴ 149 = 2n – 1

∴ 149 + 1 = 2n

∴  2n = 150

∴  n = 75

∴  149 is 75th term of A.P.

Now, We have to find sum of 75 terms i.e. S75

Sn = n/2 [2a + (n – 1) d]

∴ S75 = 75/2[2 (1) + (75 – 1) 2]

∴  S75 = 75/2 [2 + 74 [2)]

∴ S75 = 75/2 [2 + 148]

∴ S75 = 75/2 (150)

∴ S 75 = 75 (75)

∴ S75 = 5625

∴  Sum of all odd natural from 1 to 150 is 5625.