 An arithmetic sequence is one where each term can be found by adding the same number over and over again. The number could be positive or negative. Here's an example: The first term in a sequence is called t1, or more commonly a. In the example above, a = 5 The number that is being added each time is 4. This is called d, the common difference. You can find a sequence's common difference by taking any term and subtracting the term directly before it. For example, 13 - 9 = 4. It's because you subtract to get it, that d is called the common difference. For any arithmetic sequence, the term formula always looks like this:  So for the sequence above, where a = 5 and d = 4, As long as the sequence can be identified as being arithmetic, and you know a and d, you can always get the term formula for that sequence by filling a and d into tn = a + (n-1)d Let's do another example: This is an arithmetic sequence, because to get from one term to the next you always add -2. So we have a = 11, d = -2, and the term formula will be tn = a + (n-1)d The actual term formula for this sequence will be: tn = 11 + (n-1)x(-2) = 11 - 2n + 2 tn = -2n + 13 So the 30th term can now be found: t30 = -2(30) + 13 = -47 You can also use the arithmetic term formula backwards, to discover how many terms are in a particular sequence. For example:  'A church has 3 seats in the first row, 9 in the second row, and 15 in the third row. The very last row has 87 seats. How many rows of seats are in the church?' Always begin a sequence question this way: Write the sequence of numbers, and then state what you have to find. Here is the problem in simpler form: 3, 9, 15, ... 87 How many terms? To get from one term to the next, you must always add 6. So this sequence is arithmetic. a = 3 d = 6 tn = a + (n-1)d If we use 87, the last term, as tn, then n will be the number of that term. Solving for n will tell us how many terms are in the sequence up to 87. tn = a + (n-1)d 87 = 3 + (n-1) x 6 87 = 3 + 6n - 6 87 = 6n - 3 90 = 6n 15 = n So the sequence 3, 9, 15, ... 87 has 15 terms (This is the same as saying that 87 is the 15th term) There are 15 rows of seats in the church. |