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Speedy Summations
Age 16 to 18
Challenge Level 
In the video below, Alison works out \(\sum_{i=1}^{10} i\).
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Alison writes out $\sum_{i=1}^{10} i = 1+2+3+4+5+6+7+8+9+10.$
Next, Alison writes the numbers from 1 to 10, and then the same set of numbers in decreasing order, 10 to 1, underneath, then adds them in pairs. This gives $10 \times 11=110$.
Finally Alison writes the answer $55$ next to the original sum.
Next, Alison writes the numbers from 1 to 10, and then the same set of numbers in decreasing order, 10 to 1, underneath, then adds them in pairs. This gives $10 \times 11=110$.
Finally Alison writes the answer $55$ next to the original sum.
How could you adapt this method to work out the following sums?
- $\sum_{i=1}^{100} i$
- $2+4+6+\dots+96+98+100$
- $\sum_{k=1}^{20} (4k+12)$
- $37+42+47+52+\dots+102+107+112$
- The sum of the first $n$ terms of the sequence $a, (a+d), (a + 2d), (a + 3d) \dots$
After how many terms would $17+21+25+\dots$ be greater than $1000$?
Can you find the sum of all the integers less than $1000$ which are not divisible by $2$ or $3$?
Can you find a set of consecutive positive integers whose sum is 32?
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