If you take a look at sum of first n odd numbers, you can notice very interesting pattern.
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
So, sum of first n odd numbers is equal to
. Similarly, Sum of first 25 odd numbers would be equal to
.
Using the same concept, can we find sum of (1 + 3 + 5 + 7 + 9 + 11 + ……99).
We know that we have 50 odd numbers from 1 to 100. Therefore, (1 + 3 + 5 + 7 + 9 + 11 + ……99)
which is equal to sum of first 50 odd numbers.
Application:
Can you find the value of 101+103+105+107+……199 using the same concept? It becomes little tricky here.
We know that 1 + 3 + 5 + 7 + 9 + …. 199
. Here n = 100 because we know that there are 100 odd numbers from 1 and 200. (1 + 3 + 5 + 7 + 9 + …. 199 ) is the sum of first 100 odd numbers.
We also know that 1 + 3 + 5 + 7 + 9 + …. 99
(101+103+105+107+……199) = (1 + 3 + 5 + 7 + 9 + ….199) – (1 + 3 + 5 + 7 + …..99) = 10000 – 2500 = 7500
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