Question

Find the sum:  1 + 3 + 5 + 7 + ... + 199 .

Answer 1

1 + 3 + 5 + 7 + ... + 199 .

Common difference of the A.P. (d) =  `a_2 - a
_1`

= 3-1

= 2

So here,

First term (a) = 1

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 = 1 + (n-1)2

  199 = 1 + 2n - 2

199+1 = 2n 

        n = `200/2`

       n = 100

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

`S_n = 100/2 [2(1) + (100 - 1) 2 ]`

      =50 [ 2 + (99) 2]

      = 50 (2 + 198)

On further simplification, we get,

`S_n = 50(200)`

        = 10000

Therefore, the sum of the A.P is `S_n = 10000 `