Question

Find the sum of all odd numbers between 100 and 200.

Answer 1

In this problem, we need to find the sum of all odd numbers lying between 100 and 200.

So, we know that the first odd number after 0 is 101 and the last odd number before 200 is 199.

Also, all these terms will form an A.P. with the common difference of 2.

So here,

First term (a) = 101

Last term (l) = 199

Common difference (d) = 2

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

`a_n = a + (n - 1)d`

So, for the last term,

`199 = 101 + (n - 1)2`

199 = 101 + 2n - 2

199 = 99 + 2n

199 - 99 = 2n

Further simplifying,

100 = 2n

`n = 100/2`

n = 50

Now, using the formula for the sum of n terms,

`S_n = n/2 [2a + (n -1)d]`

For n = 50 we get

`S_n = 50/2 [2(101) + (50 - 1)2]`

`= 25 [202 + (49)2]`

= 25(202 + 98)

= 25(300)

= 7500

Therefore the sum of all the odd numbers lying between 100 and 200 is `S_n = 7500`