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(a) On the dotty paper, complete the next two diagrams in this sequence.
(b) (i) Complete the table.
[Table_1]
| Number of squares on the bottom row (s) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Total number of squares (T) | 1 | 3 | 6 |
(b) (ii) When the number of squares on the bottom row is 3 the total number of squares is 6. Use this information to explain how to calculate the total number of squares when there are 4 squares on the bottom row.
(c) (i) Write down the number of extra squares needed to change a pattern with 9 squares on the bottom row to one with 10 squares on the bottom row.
(c) (ii) Calculate the total number of squares when there are 10 squares on the bottom row.
(d) (i) A formula for finding the total number of squares, $T$, in terms of the number of squares on the bottom row, $s$, is
$$T = ks^2 + \frac{1}{2}s,$$
where $k$ is a constant.
Use the results in part (b)(i) to find the value of $k$.
(d) (ii) A pattern has 12 squares on the bottom row. Show that your formula in part (i) gives the correct total number of squares.
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(a) On the dotty paper, complete the next diagram in the sequence. [1] [Image_1: dot pattern]
(b) (i) Complete the table.
[Table_1: Number of squares]
Number of squares on the bottom row (s) | 1 | 2 | 3 | 4 | 5 | 6
Height \( (H) \) | 2 | 3 | 4 [1]
(b) (ii) Write down a formula for the height, \( H \), in terms of the number of squares on the bottom row, \( s \). ................................................. [1]
(c) (i) Complete the table.
[Table_2: Total number of squares]
Number of squares on the bottom row (s) | 1 | 2 | 3 | 4 | 5 | 6
Total number of squares \( (T) \) | 2 | 6 | 12 [3]
(c) (ii) Find a formula for the total number of squares, \( T \), in terms of the number of squares on the bottom row, \( s \). ................................................. [4]
(c) (iii) Find the total number of squares in a pattern with 15 squares on the bottom row. ................................................. [2]
(d) Write down a formula to calculate the number of black squares, \( N \), in a pattern with \( s \) squares on the bottom row. ................................................. [1]
(e) Calculate the number of white squares, the number of black squares and the total number of squares in a pattern with 50 squares on the bottom row.
Number of white squares = .................................................
Number of black squares = .................................................
Total number of squares = ................................................. [3]
(f) (i) A pattern of black squares and white squares has 561 black squares. Find the number of squares in the bottom row. ................................................. [3]
(f) (ii) Is it possible to have a pattern of black squares and white squares with a total of 480 squares? Give a reason for your answer.
....................... because .......................................................................... .................................................................................................................... [3]
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