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This investigation is about the lengths of spirals drawn on a square grid.
A robot starts from 0 and moves 1 unit to Corner 1.
It then turns right and moves 1 unit to Corner 2.
It then turns right and moves 2 units to Corner 3.
It then turns right and moves 2 units to Corner 4.
It then turns right and moves 3 units to Corner 5.
This forms a spiral, shown on the grid below.
The robot continues to turn and move in the same way.
(a) Continue the spiral to Corner 10. (2 marks)
(b) The length of the spiral from 0 to Corner 4 is 6 units.
Find the length of the spiral from 0 to Corner 10. (4 marks)
(c) Use your spiral to complete this table. (3 marks)
[Table_1]
587
530
512
This table shows the first five terms of a sequence.
[Table_1]
(a) For this sequence, fill in the next two terms.
(b) Write down the mathematical name for this sequence of numbers.
(c) The $n^{th}$ term for this sequence is $\frac{n(n+1)}{2}$.
Show that this is correct when $n = 5$.
560
550
573
This table shows the length, $L$, of the spiral from 0 to an \textbf{even numbered} corner, $k$.
(a) Use your table from \textit{question 1(c)} to help you complete this table.
\begin{array}{cc} \textbf{k} & \textbf{Length} \ (L) \\ \hline 2 & 2 \\ 4 & 6 \\ 6 & 12 \\ 8 & \\ 10 & \\ 12 & \\ 14 & 56 \\ 16 & \end{array}
(b) Complete this table using your answers to \textit{question 2(a)} and \textit{question 3(a)}.
\begin{array}{cccc} \textbf{n} & \textbf{Term of the sequence} & \textbf{k} & \textbf{Length} \ (L) \\ \hline 1 & 1 & 2 & 2 \\ 2 & 3 & 4 & 6 \\ 3 & 6 & 6 & 12 \\ 4 & 10 & 8 & \\ 5 & 15 & 10 & \\ 6 & & 12 & \end{array}
(i) Complete this formula for $n$ in terms of $k$:
$$n = \text{..................................................}$$
(ii) Write down the connection between the length, $L$, and the term of the sequence.
\text{...........................................................................................................................................................................}
(iii) Use \textit{part(i)}, \textit{part(ii)} and \textit{question 2(c)} to show that the formula for the length, $L$, of the spiral from 0 to an \textbf{even numbered} corner, $k$, is:
$$L = \frac{k}{2} \left( \frac{k}{2} + 1 \right).$$
(iv) Show that the formula from \textit{part(iii)} is correct for Corner 6.
(v) Show that the formula from \textit{part(iii)} is not correct when $k$ is an \textbf{odd number}.
532
516
597
(a) Write down the length of the spiral
(i) from Corner 5 to Corner 6, ...........................................................
(ii) from Corner 6 to Corner 7. ...........................................................
(b) When $k$ is an even number, find an expression, in terms of $k$, for the length of the spiral
(i) from Corner $(k - 1)$ to Corner $k$, ...........................................................
(ii) from Corner $k$ to Corner $(k + 1)$. ...........................................................
594
521
506
(a) Using question 3(b)(iii) and question 4 (b)(i), show that the length of the spiral from 0 to Corner 7 is 16 units.
(b) Find the length of the spiral from 0 to Corner 91.
540
543
562
