Mathematical induction

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Binomial coefficients

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
problem
Particularly general

Age

16 to 18

Challenge level

By proving these particular identities, prove the existence of general cases.
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Farey Fibonacci

Age

16 to 18

Challenge level

Investigate Farey sequences of ratios of Fibonacci numbers.
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Farey neighbours

Age

16 to 18

Challenge level

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?
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Tens

Age

16 to 18

Challenge level

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?
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Elevens

Age

16 to 18

Challenge level

Add powers of 3 and powers of 7 and get multiples of 11.
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Fibonacci fashion

Age

16 to 18

Challenge level

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
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Golden fractions

Age

16 to 18

Challenge level

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
problem
Water pistols

Age

16 to 18

Challenge level

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?
problem
Obviously?

Age

14 to 18

Challenge level

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.