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.