**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 = n**

^{2}......(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 = n**

^{2}[From (i)]

**∴**

**S51 = (51)**

^{2}

**∴**

**S51 = 2601**