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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?
Age 16 to 18 Challenge Level:
In the video below, Alison works out \(\sum_{i=1}^{10} i\).
How could you adapt her 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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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.