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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.
(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.
(c) Use your spiral to complete this table.
[Table_1]
575
547
588
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 $n = 5$.
566
566
554
(a) Use your table from question 1(c) to help you complete this table.
[Table_1]
| k | Length (L) |
|---|---|
| 2 | 2 |
| 4 | 6 |
| 6 | 12 |
| 8 | |
| 10 | |
| 12 | |
| 14 | 56 |
| 16 |
(b) Complete this table using your answers to question 2(a) and question 3(a).
[Table_2]
| n | Term of the sequence | k | Length (L) |
|---|---|---|---|
| 1 | 1 | 2 | 2 |
| 2 | 3 | 4 | 6 |
| 3 | 6 | 6 | 12 |
| 4 | 10 | 8 | |
| 5 | 15 | 10 | |
| 6 | 12 |
(i) Complete this formula for n in terms of k.
$$n = ext{..................................................}$$
(ii) Write down the connection between the length, L, and the term of the sequence.
...........................................................................................................
(iii) Use part (i), part (ii) and question 2(c) to show that the formula for the length, L, of the spiral from 0 to an even numbered corner, k, is
$$L = \frac{k}{2} \left(\frac{k}{2} + 1\right).$$
(iv) Show that the formula from part (iii) is correct for Corner 6.
(v) Show that the formula from part (iii) is not correct when k is an odd number.
599
561
544
(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 $(k + 1)$.
.........................................................
596
583
519
(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.
507
599
519
