Find the sum of the first n odd natural numbers. Hence find 1 + 3 + 5 + ... + 101.


5. Find the sum of the first n odd natural numbers.Hence find 1 + 3 + 5 + ... + 101.


Sol. The first n odd natural numbers are as follows: 1, 3, 5, 7, ............., n


Here, a = 1, d = t2 – t1 = 3 – 1 = 2

Sn = n/2[2a + (n – 1)d]

 Sn = n/2 [2 (1) + (n – 1) 2]

 Sn = n/2[2 + 2n – 2]

Sn = n/2 [2n]

Sn = n2 ......(i)

Now, we have,

1 + 3 + 5 + ........ + 101

Let, 101 be the nth term of A.P.

tn = 101

tn = a + (n – 1) d


 101 = a + (n – 1) d

 101 = 1 + (n – 1) 2

 101 = 1 + 2n – 2

∴  101 = 2n – 1

 101 + 1 = 2n

 2n = 102

 n = 51

101 is the 51st term of A.P.,

Now,  We have to find sum of 51 terms i.e. S51,

Sn = n2 [From (i)]


 S51 = (51)2


 S51 = 2601