Binomial theorem

problem
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Summit

Age

16 to 18

Challenge level

Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.
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Telescoping Series

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

Find the maximum value of n to the power 1/n and prove that it is a maximum.
problem
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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?
problem

Elevens

Age

16 to 18

Challenge level

Add powers of 3 and powers of 7 and get multiples of 11.
problem

Growing

Age

16 to 18

Challenge level

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)
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Remainder Hunt

Age

16 to 18

Challenge level

What are the possible remainders when the 100-th power of an integer is divided by 125?
problem

Bina-Ring

Age

16 to 18

Challenge level

Investigate powers of numbers of the form (1 + sqrt 2).
article
Binomial Coefficients

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