Arithmetic Sequences and Linear Equations Hello again. Today I would like to continue our discussion on arithmetic sequences from yesterday. I recommend that you reread yesterday’s post before you move on to this one. Here is the link: Arithmetic Sequences Recall from last week that an arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. The number d is called the common difference of the arithmetic sequence. In this post I would like to focus on the relationship between arithmetic sequences and linear equations. Arithmetic sequences and linear equations Recall the formula for the slope of a line: There is a natural correspondence between arithmetic sequences and linear equations. Each term of an arithmetic sequence can be naturally identified with a point on the corresponding line. Let’s go back to our first example of an arithmetic sequence from last week: Example 1 1, 4, 7, 10, 13, 16,… We can identify terms of the sequence with points on a line where the x-coordinate is the term number and the y-coordinate is the term itself. (1,1), (2,4), (3,7), (4,10), (5,13), (6,16),… These points all lie on the same line, and we can compute the slope of this line by using any two of these points. For example, let’s use the points (2,4) and (6,16). We have m = (16 – 4)/(6 – 2) = 12/4 = 3 Do you recognize this number? That’s right! It’s the common difference of the sequence. The identification of arithmetic sequences with linear equations gives us a nice method for finding the common difference of an arithmetic sequence as long as we know any 2 terms of the sequence. Let’s try another example. Example 2 Each term of a certain sequence is greater than the term before it. The difference between any two consecutive terms in the sequence is always the same number. If the fifth and ninth terms of the sequence are 33 and 97, respectively, what is the twelfth term? Solution: We identify the two given terms with the points (5,33) and (9,97). The common difference is then d = (97 – 33)(9 – 5) = 64/4 = 16 The twelfth term is then 97 + 16(3) = 145. The information already covered in this post and the last is all you really need to know, but for completeness, tomorrow I’m going to talk about the arithmetic sequence formula. More Problems Involving Arithmetic Sequences If you are preparing for a standardized test, take a look at the Get 800 collection of test prep books. Click on the picture below for more information. And if you liked this article, please share it with your Facebook friends: Comments comments