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