Arithmetic Sequences Hello everyone. Today I would like to talk about arithmetic sequences. Questions involving arithmetic sequences appear on the ACT, GRE and SAT math subject tests. Here is an example of an arithmetic sequence. Example 1 1, 4, 7, 10, 13, 16,… Note that the first term of this sequence is 1, the second term of this sequence is 4, and so on. So what makes this sequence arithmetic? Well notice that to get from 1 to 4 we need to add 3. To get from 4 to 7 we also add 3. To get from 7 to 10 we also add 3. In other words, the sequence is arithmetic because we always add the same number to get from any term to the next term. This special number is called the common difference of the arithmetic sequence. So why is this number called the common difference? Well another way to compute the common difference is to note that when we subtract any term from the next term we always get the same number, in this case that number is d = 3. In other words we have 4 – 1 = 3, 7 – 4 = 3, and so on. 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. Let’s try a simple example. Example 2 The second term of an arithmetic sequence is 15 and the third term is 10. What is the first term? A. –15 B. –10 C. 1/15 D. 10 E. 20 Solution: Moving backwards, to get from the third term to the second term we add 5. Therefore we add 5 more to get to the first term. So the first term is 15 + 5 = 20. This is choice E. Note that in an arithmetic sequence, you always add (or subtract) the same number to get from one term to the next. This can be done by moving forwards or backwards through the sequence. Note also that the common difference of this sequence is d = 10 – 15 = –5. Many students might mistakenly say that the common difference is 5. This particular problem was pretty simple, so we were able to solve it just by “counting.” In other words we didn’t really have to worry about the formalities of whether the common difference was positive or negative. But in harder questions we might need to be more careful. Alternate solution: Note that the terms of the sequence are getting smaller so that the first term must be larger than 15. So the answer must be choice E. In my next post we will learn a special technique that makes many seemingly difficult problems involving arithmetic sequences very easy to solve. Check it out here: Arithmetic Sequences and Linear Equations More Practice with Arithmetic Sequences If you are preparing for a standardized test, you may want to take a look at the Get 800 collection of test prep books. Click on the picture below for more information. Comments comments