### Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

### Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

### OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

# Speedy Summations

##### Age 16 to 18Challenge Level

In the video below, Alison works out $\sum_{i=1}^{10} i$.

This video has no sound

If you can't watch the video, click below for a description

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.

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?