某数列的前四项分别为 5, $\dfrac{9}{4}$, $\dfrac{13}{9}$, $\dfrac{17}{16}$。 写出该数列的第五项。 答案 ... 求该数列第 n 项 $T_n$ 关于 n 的表达式。 答案 $T_n$ = ... 以利亚（Elijah）从 his 桌游中发现了一个数字网格。 图示为该数字网格的一部分。 如图所示，一个圈出四个数字的矩形可以放置在网格的任何位置。 如果用 p 表示该矩形右上角的数字，请写出用 p 表示的该矩形左下角数字的表达式。 答案 ... 证明矩形对角线上两对数字乘积的差始终为 −7。 答案 以利亚说，矩形中四个数字之和不可能为 199。
写出相关的计算过程，证明他是正确的。 答案

1. Recognise and extend the numerator and denominator patterns to the fifth term.

The terms are
$\frac{5}{1},\ \frac{9}{4},\ \frac{13}{9},\ \frac{17}{16}.$
The numerators are $5,9,13,17$, increasing by $4$, so the next numerator is $21$.

The denominators are $1,4,9,16$, which are $1^2,2^2,3^2,4^2$, so the next denominator is $5^2=25$.

Hence the fifth term is
$\frac{21}{25}.$

1. Identify the numerator pattern and write it in terms of $n$.

The numerators are
$5, 9, 13, 17, \ldots$

They increase by $4$ each time, so the numerator of the $n$th term is
$5+4(n-1)=4n+1.$

2. Identify the denominator pattern and form the nth term.

The denominators are
$1, 4, 9, 16, \ldots$

These are square numbers:
$1^2, 2^2, 3^2, 4^2, \ldots$

So the denominator of the $n$th term is $n^2$.

Hence,
$T_n=\frac{4n+1}{n^2}.$

1. Relate the bottom left number to the top right number using the grid movement.

From the top right corner, move down one row: add $7$.

Then move left one column: subtract $1$.

So the bottom left number is
$p+7-1=p+6.$

1. Express the four corner numbers in terms of $p$.

The top right corner is $p$.

Since the top left corner is one less than $p$, it is $p-1$.

Since the bottom left corner is $p+6$ from part (b)(i), the bottom right corner is $p+7$.

So the rectangle is

2. Find and simplify the difference between the opposite-corner products.

Using one diagonal product minus the other,
$p(p+6)-(p-1)(p+7)$
$=p^2+6p-(p^2+6p-7)$
$=-7$
Therefore, the difference is always $-7$.

1. Express the four numbers in the rectangle in terms of the top right number.

Let the top right number be $p$.

Then the top left number is $p-1$, the bottom left number is $p+6$, and the bottom right number is $p+7$.

So the four numbers are
$p-1,\, p,\, p+6,\, p+7.$

2. Form and simplify the equation for the sum being 199.

If the sum were $199$,
$(p-1)+p+(p+6)+(p+7)=199$
$4p+12=199.$

3. Solve the equation and justify why the result is impossible for the grid.

$4p=187$
$p=46.75$

But $p$ represents a number in the grid, so $p$ must be an integer. Since $46.75$ is not an integer, the sum cannot be $199$.