Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 5E / b

Table of contents
- Solution b

Solution

The binomial theorem states

(x + y)^{n} = \sum_{r = 0}^{n} (r n) x^{r} y^{n - r} .

The required sum is the coefficient of

x^{n}

in

(1 - x)^{n} (1 + x)^{n} = (1 - x^{2})^{n} .

There is no

x^{n}

term when

n

is odd. When

n

is even, the relevant term has exponent

2 (n /2) = n

and coefficient

(- 1)^{n /2} (n /2 n)

. Hence

\sum_{r = 0}^{n} (- 1)^{r} (r n)^{2} = {0, (- 1)^{n /2} (n /2 n), n odd, n even .

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