    ### Linear Sequences

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

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

 N 1 2 3 4 5 Nth 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 1st term. 3 x 1 = 3. The 1st term is equal to 3 So nth term = 3n.
 (b) Work out the 14th term: 14th term = 3 x 14 = 42

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

 N 1 2 3 4 5 Nth 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 1st term. 3 x 1 = 3. But the 1st term is equal to 2. So subtract 1. Try the formula 3n − 1 for the 1st term: 1st term = 3 × 1 − 1 = 3 − 1 = 2 So nth term = 3n − 1
 (b) Work out the 16th term: 16th term = 3 × 16 − 1 = 48 − 1 = 47

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

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

 (a) Work out the first term Here the first 4 terms are missing. 1st term = 5th term − total of the first four terms 1st term = 16 − (3 × 4) = 16 − 12 = 4
 (b) Work out the formula Try the formula 3n for the 1st term. 3 x 1 = 3. But the 1st term is equal to 4. So add 1. Try the formula 3n + 1 for the 1st term: 1st term = 3 × 1 + 1 = 3 + 1 = 4 So nth term = 3n + 1
 (c) Work out the 20th term: 20th term = 3 × 20 + 1 = 60 + 1 = 61

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

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

 (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 = 34 − (6 × 5) = 34 − 30 = 4
 (b) Work out the formula Try the formula 6n for the 1st term. 6 x 1 = 6. But the 1st term is equal to 4. So subtract 2. Try the formula 6n − 2 for the 1st term: 1st term = 6 × 1 − 2 = 6 − 2 = 4 So nth term = 6n − 2
 (c) Work out the 25th term: 25th term = 6 × 25 − 2 = 150 − 2 = 148 to:  