An arithmetic sequence is a list of numbers, where each pair of consecutive numbers has the same difference (called the common difference of that sequence). For example, in the arithmetic sequence 2, 7, 12, 17, 22, the common difference is 5. Back in elementary school, when your teacher had you "count by 2s" or "count by 5s," you were actually reciting arithmetic sequences!
In Mathcounts, you may be given an arithmetic sequence with some missing terms. How would you find the unknown numbers? Usually, you should start by figuring out the common difference.
If you know the values of two consecutive numbers, lucky you! The common difference is simply equal to the difference between those two numbers. For example, if you are given the arithmetic sequence 3, 7, x, y, z, then you know that the common difference is 7-3=4. Realize that, sometimes, the common difference is a negative number. For example, in the arithmetic sequence 100, 98, 96, 94, x, the common difference is 98-100=-2.
Sometimes, however, you don't know the values of two consecutive numbers. Perhaps you are given the sequence 1, x, y, z, 9. You do know the values of two numbers (1 and 9), but they are not consecutive. To find the common difference, start by finding the total difference between the known terms: 9-1=8. This increment of 8 is reached in 4 steps:
Therefore, the difference in each of these steps is 8/4 = 2.
Now that you know the common difference, you can use it to figure out the missing number(s). In the example 1, x, y, z, 9, you've calculated that the common difference is 2. So,
x = 1 + 2 = 3
y = x + 2 = 3 + 2 = 5
z = y + 2 = 5 + 2 = 7.
Alternatively, instead of working forward from 1, you could work backward from 9. In this case, you would subtract the common difference rather than adding it:
z = 9 - 2 = 7
y = z - 2 = 7 - 2 = 5
x = y - 2 = 5 - 2 = 3
You get the same answer whether you work forward or backward, but sometimes it's faster to do one or other. For example, if the problem asked you to find the value of z in the sequence 1, x, y, z, 9, it would be faster to work backward from 9 because you wouldn't have to bother calculating the values of x or y.
TO SUMMARIZE: