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

Question

1738 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-04-17T22:41:46

$120$ $120$

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

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

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

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

The pattern continues:

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

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

Find an easy way to evaluate the following series: 3+5+7+cdots+ 17+19+213+5+7+⋯+17+19+213+5+7+cdots+ 17+19+21?