Band 1 – Page 2

Jay is making a pattern using triangular tiles. The pattern has 3 tiles in the first row, 5 tiles in the second row, and each successive row has 2 more tiles than the previous row.

How many tiles would Jay use in row 20? (2 marks)

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How many tiles would Jay use altogether to make the first 20 rows? (1 mark)

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Jay has only 200 tiles. How many complete rows of the pattern can Jay make? (2 marks)

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Show Answers Only

`\ 41`
`440`
`13\ text(rows)`

Show Worked Solutions

i.	`T_1`	`=a=3`
	`T_2`	`=a+d=5`
	`T_3`	`=a+2d=7`

`=>\ text(AP where)\ \ a=3,\ \ d=2`

`\ \ \ \ \ vdots`

`T_20`	`=a+19d`
	`=3+19(2)`
	`=41`

`:.\ text(Row 20 has 41 tiles.)`

MARKER’S COMMENT: Better responses stated the formula BEFORE any calculations were performed. This enabled students to get some marks if they made an error in their working.

ii. `S_20`	`=\ text(the total number of tiles in first 20 rows)`
`S_20`	`=n/2(a+l)`
	`=20/2(3+41)`
	`=440`

`:.\ text(There are 440 tiles in the first 20 rows.)`

iii. `text(If Jay only has 200 tiles, then)\ \ S_n<=200`

NOTE: Examiners often ask questions requiring `n` to be found using the formula `S_n=n/2[2a+(n-1)d]` as this requires the solving of a quadratic, and interpretation of the answer.

`n/2(2a+(n-1)d)`	`<=200`
`n/2(6+2n-2)`	`<=200`
`n(n+2)`	`<=200`
`n^2+2n-200`	`<=0`

`n`	`=(-2+-sqrt(4+41200))/(2*1)`
	`=(-2+-sqrt804)/2`
	`=-1+-sqrt201`
	`=13.16\ \ text{(answer must be positive)}`

`:.\ text(Jay can complete 13 rows.)`