### Answers − Linear Sequences

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

N | 1 | 2 | 3 | 4 | 5 | |||||

N^{th} 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 1^{st} term. | |

7 x 1 = 7. The 1^{st} term is equal to 7 | |

So n^{th} term = 7n. |

(b) | Work out the 55^{th} term: |

55^{th} 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 | |||||

N^{th} 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 1^{st} term. | |

7 x 1 = 7. But the 1^{st} term is equal to 11. So add 4. | |

Try the formula 7n + 4 for the 1^{st} term: | |

1^{st} term = 7 × 1 + 4 = 7 + 4 = 11 | |

So n^{th} term = 7n + 4 |

(b) | Work out the 20^{th} term: |

20^{th} 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 | |||||

N^{th} term | ~ | 37 | 44 | 51 | ||||||

Difference | +7 | +7 |

(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 = 37 − (7 × 5) = 37 − 35 = 2 |

(b) | Work out the formula |

Try the formula 7n for the 1^{st} term. | |

7 x 1 = 7. But the 1^{st} term is equal to 2. So subtract 5. | |

Try the formula 7n − 5 for the 1^{st} term: | |

1^{st} term = 7 × 1 − 5 = 7 − 5 = 2 | |

So n^{th} term = 7n − 5 |

(c) | Work out the 17^{th} term: |

17^{th} 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 | |||||

N^{th} 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 1^{st} term. | |

14 x 1 = 14. The 1^{st} term is equal to 14 | |

So n^{th} term = 14n. |

(b) | Work out the 29^{th} term: |

29^{th} 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 | |||||

N^{th} 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 1^{st} term. | |

14 x 1 = 14. But the 1^{st} term is equal to 5. So subtract 9. | |

Try the formula 14n − 9 for the 1^{st} term: | |

1^{st} term = 14 × 1 − 9 = 14 − 9 = 5 | |

So n^{th} term = 14n − 9 |

(b) | Work out the 17^{th} term: |

17^{th} 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 | |||||

N^{th} term | ~ | 110 | 124 | 138 | ||||||

Difference | +14 | +14 |

(a) | Work out the first term |

Here the first 6 terms are missing. | |

1^{st} term = 7^{th} term − total of the first six terms | |

1^{st} term = 110 − (6 × 14) = 110 − 84 = 26 |

(b) | Work out the formula |

Try the formula 14n for the 1^{st} term. | |

14 x 1 = 14. But the 1^{st} term is equal to 26. So add 12. | |

Try the formula 14n + 12 for the 1^{st} term: | |

1^{st} term = 14 × 1 + 12 = 14 + 12 = 26 | |

So n^{th} term = 14n + 12 |

(c) | Work out the 15^{th} term: |

15^{th} 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 | |||||

N^{th} term | ~ | 81 | 92 | 103 | ||||||

Difference | +11 | +11 |

(a) | Work out the first term |

Here the first 7 terms are missing. | |

1^{st} term = 8^{th} term − total of the first seven terms | |

1^{st} term = 81 − (7 × 11) = 81 − 77 = 4 |

(b) | Work out the formula |

Try the formula 11n for the 1^{st} term. | |

11 x 1 = 11. But the 1^{st} term is equal to 4. So subtract 7. | |

Try the formula 11n − 7 for the 1^{st} term: | |

1^{st} term = 11 × 1 − 7 = 11 − 7 = 4 | |

So n^{th} term = 11n − 7 |

(c) | Work out the 24^{th} term: |

24^{th} term = 11 × 24 − 7 = 264 − 7 = 257 |

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