Binomial theorem

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

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

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

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

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

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

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

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

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