1. When the expression $\frac{\left(2^{2}-1\right) \times\left(3^{2}-1\right) \times\left(4^{2}-1\right) \times\left(5^{2}-1\right)}{(2 \times 3) \times(3 \times 4) \times(4 \times 5) \times(5 \times 6)}$ is simplified, which of the following is obtained?
A $\frac{1}{2}$
B $\frac{1}{3}$
C $\frac{1}{4}$
D $\frac{1}{5}$
E $\frac{1}{6}$
Solution
D
Using the standard factorization $n^{2}-1=(n-1)(n+1)$, we have
$$
\begin{aligned}
\frac{\left(2^{2}-1\right) \times\left(3^{2}-1\right) \times\left(4^{2}-1\right) \times\left(5^{2}-1\right)}{(2 \times 3) \times(3 \times 4) \times(4 \times 5) \times(5 \times 6)} & =\frac{(1 \times 3) \times(2 \times 4) \times(3 \times 5) \times(4 \times 6)}{(2 \times 3) \times(3 \times 4) \times(4 \times 5) \times(5 \times 6)} \\
& =\frac{1 \times 2 \times 3 \times 4}{2 \times 3 \times 4 \times 5} \\
& =\frac{1}{5} .
\end{aligned}
$$

For inVestigation
1.1 Write the following expressions in simplified form.
(a) $\frac{\left(2^{2}-1\right) \times\left(3^{2}-1\right) \times\left(4^{2}-1\right) \times\left(5^{2}-1\right) \times\left(6^{2}-1\right)}{(2 \times 3) \times(3 \times 4) \times(4 \times 5) \times(5 \times 6) \times(6 \times 7)}$,
(b) $\frac{\left(2^{2}-1^{2}\right) \times\left(3^{2}-2^{2}\right) \times\left(4^{2}-3^{2}\right) \times\left(5^{2}-4^{2}\right)}{1 \times 3 \times 5 \times 7}$.
(c) $\frac{\left(3^{4}-1\right) \times\left(4^{4}-1\right) \times\left(5^{4}-1\right) \times\left(6^{4}-1\right)}{(2 \times 4) \times(3 \times 5) \times(4 \times 6) \times(5 \times 7)}$.