# How do you add all the odd numbers between 1-99 inclusive?

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Form a sequence

This is the sequence of all the odd numbers between 1 and 99, endpoints included.

Clearly this is an arithmetic sequence with common difference d = 2 between terms.

The general term for this sequence may be given as :

Hence,

Writing

Now to add all the terms of this sequence, yields the arithmetic series

From the formula for sum of an arithmetic series with constant difference d, we obtain the solution as

Or, using the other formula if the first and last terms are known is:

sum

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More simply, considering the sum of the series

#1 + 3 + 5 + 7 + ...+ 93 + 95 + 97 + 99#

Note that the numbers may be paired off

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