In section 10.2 we dealt with sequences where the differences between the terms was a constant value. In this section we extend this idea to sequences where the differences are *not constant*.

(a)

Calculate the first 6 terms of the sequence defined by the quadratic formula,

u_n = n^2 + n – 1

Substituting u_{1} | = 1^{2} + 1 – 1 |

= 1 |

For n = 2, | u_{2} | = 2^{2} + 2 – 1 |

= 5 |

For n = 3, | u_{3} | = 3^{2} + 3 – 1 |

= 11 |

For n = 4, | u_{4} | = 4^{2} + 4 – 1 |

= 19 |

For n = 5, | u_{5} | = 5^{2} + 5 – 1 |

= 29 |

For n = 6, | u_{6} | = 6^{2} + 6 – 1 |

= 41 |

So the first 6 terms are,

1, 5, 11, 19, 29, 41

(b)

Calculate the first differences between the terms.

The differences can now be calculated,

1 | 5 | 11 | 19 | 29 | 41 | ... |

4 | 6 | 8 | 10 | 12 |

(c)

Comment on the results you obtain.

Note that the differences between the first differences are constant. They are all equal to 2. These are called the *second differences*, as shown below.

Sequence | 1 | 5 | 11 | 19 | 29 | 41 | ... |
---|

First differences | 4 | 6 | 8 | 10 | 12 |
---|

Second differences | 2 | 2 | 2 | 2 |
---|

Calculate the first 5 terms of the sequence defined by the quadratic formula

(a)

Calculate the first 6 terms of the sequence defined by the quadratic formula,

u_n = 3n^2 – n – 2

For n = 1, | u_{1} | = 3 × 1^{2} – 1 – 2 |

= 3 – 1 – 2 | ||

= 0 |

For n = 2, | u_{2} | = 3 × 2^{2} – 2 – 2 |

= 8 |

For n = 3, | u_{3} | = 3 × 3^{2} – 3 – 2 |

= 22 |

For n = 4, | u_{4} | = 3 × 4^{2} – 4 – 2 |

= 42 |

For n = 5, | u_{5} | = 3 × 5^{2} – 5 – 2 |

= 68 |

So the sequence is,

2, 8, 22, 42, 68, ...

(b)

Determine the first and second differences for this sequence.

The differences can now be calculated,

Sequence | 0 | 8 | 22 | 42 | 68 | ... |
---|

First differences | 8 | 14 | 20 | 26 |
---|

Second differences | 6 | 6 | 6 |
---|

(c)

Comment on your results.

Again, the second differences are constant; this time they are all 6.

Note: For a sequence defined by a *quadratic formula*, the second differences will be constant and equal to twice the number of *n*^{2} .

For example,

u_n = n^2 + n – 1 Second difference = 2u_n = 3n^2 – n – 2 Second difference = 6

u_n = 5n^2 – n + 7 Second difference = 10

Determine a formula for the general term of the sequence,

2, 9, 20, 35, 54, ...

Consider the first and second differences of the sequence:

2 | 9 | 20 | 35 | 54 | ... |

7 | 11 | 15 | 19 |

4 | 4 | 4 |

As the second differences are constant and equal to 4, the formula will begin

u_n = 2n^2 + ...

To determine the rest of the formula, subtract 2n^2 from each term of the sequence, as shown below:

Sequence | 2 | 9 | 20 | 35 | 54 | ... |
---|---|---|---|---|---|---|

2n^2 | 2 | 8 | 18 | 32 | 50 |

New sequence | 0 | 1 | 2 | 3 | 4 |
---|

1 | 1 | 1 | 1 |

The new sequence has a constant difference of 1 and begins with 0, so for this sequence the formula is *n* – 1.

Combining this with the 2n^2 gives

u_n = 2n^2 + n – 1

(a)

Calculate the first and second differences for the sequence,

4, 1, 0, 1, 4, 9, ...

4 | 1 | 0 | 1 | 4 | 9 | ... |

-3 | -1 | 1 | 3 | 5 |

2 | 2 | 2 | 2 |

(b)

Use the differences to determine the next 2 terms of the sequence.

Extending the sequences above gives,

4 | 1 | 0 | 1 | 4 | 9 | 16 | 25 | ... |

-3 | -1 | 1 | 3 | 5 | 7 | 9 |

2 | 2 | 2 | 2 | 2 | 2 |

(c)

Determine a formula for the general term of the sequence.

As the second differences are constant and all equal to 2, the formula will contain an '*n*^{2}' term, and be of the form

u_n = n^2 + *a*n + *b*

We must now determine the values of *a* and *b*. The easiest way to do this is to subtract *n*^{2} from each term of the sequence, to form a new, simpler sequence.

our sequence | 4 | 1 | 0 | 1 | 4 | 9 |
---|---|---|---|---|---|---|

sequence n^{2} | 1 | 4 | 9 | 16 | 25 | 36 |

new sequence | 3 | –3 | –9 | –15 | –21 | –27 |

The new sequence

3 | –3 | –9 | –15 | –21 | –27 | ... |

–6 | –6 | –6 | –6 | –6 |

has constant first differences of –6 so will be given by –6*n* + *b*.

Using the first term gives,

3 | = –6 × 1 + b |

b | = 9 |

Thus the formula for the simpler sequence is –6*n* + 9.

Now combining this with the *n*^{2} term gives,

u_n = n^2 – 6n + 9

Note: To type indeces on this page use ^ sign. e.g.n^2 :