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# Slick Summing

**Notes and Background**

If you enjoyed this problem you may be interested to read the article Clever Carl, which tells the story of the young Gauss working on sums like the ones in this problem. You may also be interested in this video which shows a dozen proofs of the formula of the sum of the first n positive integers.
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### Picture Story

### Sums of Powers - A Festive Story

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Age 14 to 16

Challenge Level

*Slick Summing printable worksheet*

In the video below, Charlie works out $1+2+3+4+5+6+7+8+9+10$.

Can you see how his method works?

How could you adapt his method to work out the following sums?

$1 + 2 + 3 + \dots + 19 + 20$

$1 + 2 + 3 + \dots + 99 + 100$

$40 + 41 + 42 + \dots + 99 + 100$

Can Charlie's method be adapted to sum sequences that don't go up in ones?

$1 + 3 + 5 + \dots + 17 + 19$

$2 + 4 + 6 + \dots + 18 + 20$

$42 + 44 + 46 + \dots + 98 + 100$

Can you find an expression for the following sum?

$1 + 2 + 3 + \dots + (n - 1) + n$

If you enjoyed this problem you may be interested to read the article Clever Carl, which tells the story of the young Gauss working on sums like the ones in this problem. You may also be interested in this video which shows a dozen proofs of the formula of the sum of the first n positive integers.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

A story for students about adding powers of integers - with a festive twist.