Find an easy way to evaluate the following series: #3+5+7+cdots+ 17+19+21#?

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Answer 1 · 2016-04-17T22:41:46

#120#

Note that as you work inwards, groups of #24# form.

Add the first and last numbers, #3# and #21#, to get #24#.

Since the next two numbers inwards are just #2# less and greater than the numbers we added before, their sum will also be #24#. That is:

#5+19=(3+2)+(21-2)=(3+21)+(2-2)=24#

The pattern continues:

#{(3+21=24),(5+19=24),(7+17=24),(9+15=24),(11+13=24):}#

Thus, the sum is #5xx24=120#.