Volume 33
, Issue 1

1997

Welcome (belatedly) to a new year – I don’t know what life is like for you, but with the present cut-backs, it is getting harder and harder to have Parabola out on time.

In a previous article (Parabola, Volume 32 Number 2), the swing and reverse swing of a cricket ball was discussed.

In the 3-Unit Maths course you are asked to prove (by induction) various formulae such as
$$ 1^2+2^2+3^2+.....+n^2 = \sum _{x=1}^n x^2 = \frac{1}{6}n(n+1)(2n+1),$$

When you began to sketch curves early in high school, you evaluated the “$y$-value” for several “$x$-values”, plotted the resulting points and then joined them up as smoothly as you could.

Q993 Consider
\begin{eqnarray*}
p(n)&=&a_0+a_1n+a_2n^2+\cdots+a_kn^k\ ,\\
  q(n)&=&b_0+b_1\binom{n}{1}+b_2\binom{n}{2}+\cdots+b_k\binom{n}{k}\ ,
\end{eqnarray*}

Q.985 For what values of the positive integer $n$ is

  1. $5n + 2$
  2. $7n + 2$

a perfect square?