DevelopMaths.com High School Maths

Linear Sequences

This is where the difference between each number in the sequence is the same

Example 1. Find the 14^th term in the sequence 3, 6, 9, 12, 15, ...

N		1		2		3		4		5
N^th term		3		6		9		12		15
Difference				+3		+3		+3		+3

The difference between each term is 3, so the formula involves the 3 times table.

(a)	Work out the formula
	Try the formula 3n for the 1^st term.
	3 x 1 = 3. The 1^st term is equal to 3
	So n^th term = 3n.

(b)	Work out the 14^th term:
	14^th term = 3 x 14 = 42

Example 2. Find the 16^th term in the sequence 2, 5, 8, 11, 14, ...

N		1		2		3		4		5
N^th term		2		5		8		11		14
Difference				+3		+3		+3		+3

The difference between each term is 3, so the formula involves the 3 times table.

(a)	Work out the formula
	Try the formula 3n for the 1^st term.
	3 x 1 = 3. But the 1^st term is equal to 2. So subtract 1.
	Try the formula 3n − 1 for the 1^st term:
	1^st term = 3 × 1 − 1 = 3 − 1 = 2
	So n^th term = 3n − 1

(b)	Work out the 16^th term:
	16^th term = 3 × 16 − 1 = 48 − 1 = 47

Example 3. The 5^th, 6^th and 7^th term in a sequence are 16, 19 and 20. Work out the 1^st and 20^th term

N		1		~		5		6		7
N^th term				~		16		19		22
Difference								+3		+3

(a)	Work out the first term
	Here the first 4 terms are missing.
	1^st term = 5^th term − total of the first four terms
	1^st term = 16 − (3 × 4) = 16 − 12 = 4

(b)	Work out the formula
	Try the formula 3n for the 1^st term.
	3 x 1 = 3. But the 1^st term is equal to 4. So add 1.
	Try the formula 3n + 1 for the 1^st term:
	1^st term = 3 × 1 + 1 = 3 + 1 = 4
	So n^th term = 3n + 1

(c)	Work out the 20^th term:
	20^th term = 3 × 20 + 1 = 60 + 1 = 61

Example 4. The 6^th, 7^th and 8^th term in a sequence are 34, 40 and 46. Work out the 1^st and 25^th term

N		1		~		6		7		8
N^th term				~		34		40		46
Difference								+6		+6

(a)	Work out the first term
	Here the first 5 terms are missing.
	1^st term = 6^th term − total of the first five terms
	1^st term = 34 − (6 × 5) = 34 − 30 = 4

(b)	Work out the formula
	Try the formula 6n for the 1^st term.
	6 x 1 = 6. But the 1^st term is equal to 4. So subtract 2.
	Try the formula 6n − 2 for the 1^st term:
	1^st term = 6 × 1 − 2 = 6 − 2 = 4
	So n^th term = 6n − 2

(c)	Work out the 25^th term:
	25^th term = 6 × 25 − 2 = 150 − 2 = 148

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