# Find the Sum: 1 + 3 + 5 + 7 + ... + 199 . - Mathematics

Sum

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

#### Solution

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

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Exercise 5.6 | Q 13.4 | Page 51