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