The diagram shows a sequence of square grids made from matchsticks. Pattern \(n\) is an \(n\) by \(n\) grid of small squares. Pattern number 1 2 3 4 Number ...

Pattern \(n\) is an \(n\) by \(n\) grid of small squares, so the number of small squares is a square number, while the matchsticks form the horizontal and vertical lines of the grid.

(a) For Pattern \(4\): small squares \(=4^{2}=16\) [B1], and matchsticks \(=40\) [B1], continuing the pattern \(4,12,24,40\) whose differences are \(8,12,16\).

(b) An \(n\) by \(n\) grid holds \(n^{2}\) small squares [B1].

(c) The matchstick counts have first differences \(8,12,16\) and a constant second difference of \(4\), so the rule is quadratic with \(n^{2}\) coefficient \(\frac{4}{2}=2\), giving \(2n^{2}\) [M1]. Subtracting \(2n^{2}=2,8,18,32\) from \(4,12,24,40\) leaves \(2,4,6,8\), which is \(2n\) [M1]. Hence

Matchsticks \(=2n^{2}+2n\) [A1] oe, which factorises as \(2n(n+1)\)

The structure of the grid confirms this: there are \(n+1\) horizontal lines each made of \(n\) matches, and \(n+1\) vertical lines each made of \(n\) matches, giving \(2n(n+1)\).

(d) Set the expression equal to \(220\):

1. \(2n^{2}+2n=220\) [M1]
2. Divide by \(2\) and rearrange: \(n^{2}+n-110=0\), which factorises as \((n+11)(n-10)=0\) [M1] oe, since \(11\times(-10)=-110\) and \(11-10=1\).
3. The roots are \(n=-11\) and \(n=10\). A pattern number cannot be negative, so \(n=10\) [A1]

Check: \(2(100)+2(10)=200+20=220\). Rejecting the negative root by referring to the context is expected in pattern questions, since only positive whole numbers make sense as pattern numbers.