Answers − Linear Sequences

1.   Work out the 55^th term in the sequence 7, 14, 21, 28, 35

N		1		2		3		4		5
Nth term		7		14		21		28		35
Difference				+7		+7		+7		+7

The difference between each term is 7, so the formula involves the 7 times table.

(a)	Work out the formula
	Try the formula 7n for the 1st term.
	7 x 1 = 7. The 1st term is equal to 7
	So nth term = 7n.

(b)	Work out the 55th term:
	55th term = 7 x 55 = 385

2.   Work out the 20^th term in the sequence 11, 18, 25, 32, 39

N		1		2		3		4		5
Nth term		11		18		25		32		39
Difference				+7		+7		+7		+7

The difference between each term is 7, so the formula involves the 7 times table.

(a)	Work out the formula
	Try the formula 7n for the 1st term.
	7 x 1 = 7. But the 1st term is equal to 11. So add 4.
	Try the formula 7n + 4 for the 1st term:
	1st term = 7 × 1 + 4 = 7 + 4 = 11
	So nth term = 7n + 4

(b)	Work out the 20th term:
	20th term = 7 × 20 + 4 = 140 + 4 = 144

3.   The 6^th, 7^th and 8^th terms in a sequence are 37, 44 and 51. Work out the 1^st and 17^th terms

N		1		~		6		7		8
Nth term				~		37		44		51
Difference								+7		+7

(a)	Work out the first term
	Here the first 5 terms are missing.
	1st term = 6th term − total of the first five terms
	1st term = 37 − (7 × 5) = 37 − 35 = 2

(b)	Work out the formula
	Try the formula 7n for the 1st term.
	7 x 1 = 7. But the 1st term is equal to 2. So subtract 5.
	Try the formula 7n − 5 for the 1st term:
	1st term = 7 × 1 − 5 = 7 − 5 = 2
	So nth term = 7n − 5

(c)	Work out the 17th term:
	17th term = 7 × 17 − 5 = 119 − 5 = 114

4.   Work out the 29^th term in the sequence 14, 28, 42, 56, 70

N		1		2		3		4		5
Nth term		14		28		42		56		70
Difference				+14		+14		+14		+14

The difference between each term is 14, so the formula involves the 14 times table.

(a)	Work out the formula
	Try the formula 14n for the 1st term.
	14 x 1 = 14. The 1st term is equal to 14
	So nth term = 14n.

(b)	Work out the 29th term:
	29th term = 14 x 29 = 406

5.   Work out the 17^th term in the sequence 5, 19, 33, 47, 61

N		1		2		3		4		5
Nth term		5		19		33		47		61
Difference				+14		+14		+14		+14

The difference between each term is 14, so the formula involves the 14

(a)	Work out the formula
	Try the formula 14n for the 1st term.
	14 x 1 = 14. But the 1st term is equal to 5. So subtract 9.
	Try the formula 14n − 9 for the 1st term:
	1st term = 14 × 1 − 9 = 14 − 9 = 5
	So nth term = 14n − 9

(b)	Work out the 17th term:
	17th term = 14 × 17 − 9 = 238 − 9 = 229

6.   The 7^th, 8^th and 9^th terms in a sequence are 110, 124 and 138. Work out the 1^st and 15^th term

N		1		~		7		8		9
Nth term				~		110		124		138
Difference								+14		+14

(a)	Work out the first term
	Here the first 6 terms are missing.
	1st term = 7th term − total of the first six terms
	1st term = 110 − (6 × 14) = 110 − 84 = 26

(b)	Work out the formula
	Try the formula 14n for the 1st term.
	14 x 1 = 14. But the 1st term is equal to 26. So add 12.
	Try the formula 14n + 12 for the 1st term:
	1st term = 14 × 1 + 12 = 14 + 12 = 26
	So nth term = 14n + 12

(c)	Work out the 15th term:
	15th term = 14 × 15 + 12 = 210 + 12 = 222

7.   The 8^th, 9^th and 10^th terms in a sequence are 81, 92 and 103. Work out the 1^st and 24^th term

N		1		~		8		9		10
Nth term				~		81		92		103
Difference								+11		+11

(a)	Work out the first term
	Here the first 7 terms are missing.
	1st term = 8th term − total of the first seven terms
	1st term = 81 − (7 × 11) = 81 − 77 = 4

(b)	Work out the formula
	Try the formula 11n for the 1st term.
	11 x 1 = 11. But the 1st term is equal to 4. So subtract 7.
	Try the formula 11n − 7 for the 1st term:
	1st term = 11 × 1 − 7 = 11 − 7 = 4
	So nth term = 11n − 7

(c)	Work out the 24th term:
	24th term = 11 × 24 − 7 = 264 − 7 = 257

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