I’ve added a script which helps to understand arithmetic sequences.

At a previous job of mine, there was a policy of holding a dinner party for the company each time we hired a new employee. At these dinners, each employee was treated to a $20 dinner at the expense of the company. There was also a manager responsible for keeping track of the costs of these dinners.

In computing the costs, the manager noticed that each time there is a new dinner, it was $20 more expensive than the last one. So if we let a_{1} represent the cost of the first dinner, and let a_{i} represent the cost of the i^{th} dinner, then we see that a_{i} = a_{i-1} + 20. Sequences like this, where t arise quite often in practice and are called arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any two consecutive numbers is constant.

For the example above, the term a_{n} will represent the cost of dinner after the n^{th} employee has joined the company (assuming that no employees have left the company over this time period). Also the term S_{n} will represent the total cost the company has paid towards these dinners.

Before we continue with this example, consider the following table which lists the first five terms of an arithmetic sequence as well as the common difference and the first five sums of this sequence.

term number | term value | diff | sum number | sum value |

a_{1} |
4 | 3 | S_{1} |
4 |

a_{2} |
7 | 3 | S_{2} |
11 |

a_{3} |
10 | 3 | S_{3} |
21 |

a_{4} |
13 | 3 | S_{4} |
34 |

a_{5} |
16 | 3 | S_{5} |
50 |

One of the beauties of arithmetic sequences is that if we know the first term (a_{1}) and the common difference (d), then we can easily calculate the terms a_{n} and S_{n} for any n with the following formulas:

a_{n} = a_{1} + d*(n – 1), where d is the common difference.

S_{n} = n*(a_{1} + a_{n})/2

We can use these formulas to derive more information about the sequence. For example, if my manager wanted to estimate the cost of dinners once we had added 30 new employees, this would be term a_{30} of the sequence, which we can evaluate with the above formula by a_{30} = a_{1} + d*(n – 1) = 0 + 20*(30 – 1) = 0 + 20 * 29 = 580.

The script is available at http://www.learninglover.com/examples.php?id=33.

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I’m impressed, I must say.