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Binomial Coefficients
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
Binomial theorem
problem
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
Bina-ring
Age
16 to 18
Challenge level
Investigate powers of numbers of the form (1 + sqrt 2).
problem
Elevens
Age
16 to 18
Challenge level
Add powers of 3 and powers of 7 and get multiples of 11.
problem
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
Summit
Age
16 to 18
Challenge level
Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.
problem
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
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
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)
problem
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 $S_1, S_2, ... S_{r-1}$.