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Binomial

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Powerful Factors

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

Particularly General

Age 16 to 18 Challenge Level:

Expand out the brackets in a systematic, organised manner.

 

You will need to use a double angle trig formula and difference of two squares. Considering $n=1$ might be useful.

 

There is perhaps no need to provide a proof for your identities if you are very clear in your mind how such a proof would be constructed, although you might wish to produce a proof if interested.