- Write down the next three members of the sequence of numbers\[1,4,9,16,\dotsc\]
- Calculate the 300th member of the sequence.

The value of each member is simply the square of its position. So the 300th member of the sequence is given by \(300^2=90\ 000.\)

- Calculate the first five members of the sequence of differences\[4-1\ ,9-4\ ,16-9,\dotsc\]
- If \(a\),\(b\),\(c\), are
*consecutive*members of the sequence in (i) write down a formula connecting \(c-b\) and \(b-a\). Rewrite this formula so \(b\) is the subject.

The first five members of the sequence are \(4-1,9-4,16-9,25-16,36-25,\) or \(3,5,7,9,11.\)

If \(a\),\(b\) and \(c\) are consecutive members of the initial sequence, \(b-a\) and \(c-b\) are consecutive members of the sequence of differences.

This means \(c-b\) is \(2\) larger than \(b-a\), or \(c-b=b-a+2,\) which gives \(b=\frac{1}{2} \left(a+c \right) -1\).

- Use the result of (iv) and the fact that \[ 8677^2 = 75290329 \qquad \text{ and} \qquad 8679^2 = 75325041\] to calculate \(8678^2\) making your method clear.

The numbers \(8677\), \(8678\) and \(8679\) are three consecutive integers, and we know the squares of the first and last of these.

This gives us our \(a\) and \(c\). Now we can substitute those numbers into the equation and obtain

\[\begin{align*} b &= \frac{1}{2}(a + c) -1 \\ 8678^2 &= \frac{1}{2} \left(75290329+75325041 \right) -1 \\ &=75307584. \end{align*}\]