### 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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