Telescoping Series

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

Growing

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

Remainder Hunt

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

Binomial

Age 16 to 18 Challenge Level:

Well done Ang Zhi Ping from River Valley High School, Singapore for your excellent solution to this question.

Let the binomial coefficient $n!/r!(n-r)!$ be denoted by $${n\choose r}.$$

By considering powers of $(1 + x)$ show that $\sum_{k=0}^n {n\choose k}^2 = {2n \choose n}$

As $(1 + x)^n(1 + x)^n = (1 + x)^{2n} \quad (1)$, we write down the Binomial expansion giving: \begin{equation*}\left[\sum_{p=0}^n {n\choose p} x^p\right]\left[\sum_{q=0}^n {n\choose q} x^q\right] = \sum_{r=0}^{2n} {2n\choose r}x^r. \end{equation*} The left hand side of the equation is \begin{equation*}\left[{n\choose 0} + {n\choose 1}x + \cdots + {n\choose n}x^n\right] \left[{n\choose 0} + {n\choose 1}x + \cdots + {n\choose n}x^n\right].\end{equation*} So the coefficient of $x^n$ on the left hand side of (1) is \begin{equation*}{n\choose 0}{n\choose n} + {n\choose 1}{n\choose n-1} + {n\choose 2}{n\choose n-2} + \cdots +{n\choose n-1}{n\choose 1} + {n\choose n}{n\choose 0}.\end{equation*} Since $${n\choose r} = {n\choose n-r}$$ we see that the coefficient of $x^n$ on the left hand side of (1) is $$\sum_{k=0}^n {n\choose k}^2.$$ As the coefficient of $x^n$ on the right hand side of (1) is $${2n\choose n}$$ the given formula is proven.