How to Find Any Term of an Arithmetic Sequence

Step-by-Step Guide

1. 1

   Step 1: Find the common difference for the sequence.

   When you are presented with a list of numbers, you may be told that the list is an arithmetic sequence, or you may need to figure that out for yourself.
   
   The first step is the same in either case.
   
   Select the first two consecutive terms in the list.
   
   Subtract the first term from the second term.
   
   The result is the common difference of your sequence.For example, suppose you have the list 1,4,7,10,13{\displaystyle 1,4,7,10,13}....
   
   Subtract 4−1{\displaystyle 4-1} to find the common difference of
   3.
   
   Suppose you have a list of terms that decreases, such as 25,21,17,13{\displaystyle 25,21,17,13}….
   
   You still subtract the first term from the second to find the difference.
   
   In this case, that gives you 21−25=−4{\displaystyle 21-25=-4}.
   
   The negative result means that your list is decreasing as you read from left to right.
   
   You should always check that the sign of the difference matches the direction that the numbers seem to be going.
2. 2

   Step 2: Check that the common difference is consistent.

   Finding the common difference for just the first two terms does not ensure that your list is an arithmetic sequence.
   
   You need to make sure that the difference is consistent for the whole list.
   
   Check the difference by subtracting two different consecutive terms in the list.
   
   If the result is consistent for one or two other pairs of terms, then you probably have an arithmetic sequence.
   
   Working with the same example, 1,4,7,10,13{\displaystyle 1,4,7,10,13}… choose the second and third terms of the list.
   
   Subtract 7−4{\displaystyle 7-4}, and you find that the difference is still
   3.
   
   To confirm, check one more example and subtract 13−10{\displaystyle 13-10}, and you find that the difference is consistently
   3.
   
   You can be pretty sure that you are working with an arithmetic sequence.
   
   It is possible for a list of numbers to appear to be an arithmetic sequence based on the first few terms, but then fail after that.
   
   For example, consider the list 1,2,3,6,9{\displaystyle 1,2,3,6,9}….
   
   The difference between the first and second terms is 1, and the difference between the second and third terms is also
   1.
   
   However, the difference between the third and fourth terms is
   3.
   
   Because the difference is not common for the entire list, then this is not an arithmetic sequence. , Finding the next term of an arithmetic sequence after you know the common difference is easy.
   
   Simply add the common difference to the last term of the list, and you will get the next number.
   
   For example, in the example of 1,4,7,10,13{\displaystyle 1,4,7,10,13}…, to find the next number in the list, add the common difference of 3 to the last given term.
   
   Adding 13+3{\displaystyle 13+3} results in 16, which is the next term.
   
   You can continue adding 3 to make your list as long as you like.
   
   For example, the list would be 1,4,7,10,13,16,19,22,25{\displaystyle 1,4,7,10,13,16,19,22,25}….
   
   You can do this as long as you like.
3. 3

   Step 3: Add the common difference to the last given term.

Detailed Guide

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