The first four terms of a sequence are 5, $\dfrac{9}{4}$, $\dfrac{13}{9}$, $\dfrac{17}{16}$. State the fifth term of the sequence. Answer ... Find an expression, in terms of n, for the nth term, $T_n$, of this sequence. Answer $T_n$ = ... Elijah finds a number grid from his board game. The diagram shows part of a number grid. A rectangle outlining four numbers, as shown, can be placed anywhere on the grid. If p represents the number in the top right corner of the rectangle, write down an expression, in terms of p, for the number in the bottom left corner of the rectangle. Answer ... Show that the difference between the products of the numbers in the opposite corners of the rectangle is always −7 . Answer Elijah says it is impossible for the sum of the four numbers in the rectangle to be 199.
Justify with relevant working, why he is correct. Answer

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 $199$.