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.

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.

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.

For example, suppose you have the list 0,4{\displaystyle 0,4},___,12,16,20{\displaystyle 12,16,20}…. Start by subtracting 4−0{\displaystyle 4-0} to find a difference of 4. Check this against two other consecutive terms, such as 16−12{\displaystyle 16-12}. The difference is again 4. You can proceed.

In our working example, 0,4{\displaystyle 0,4},____,12,16,20{\displaystyle 12,16,20}…, the term preceding the space is 4, and our common difference for this list is also 4. So add 4+4{\displaystyle 4+4} to get 8, which should be the number in the blank space.

In the working example, 0,4{\displaystyle 0,4},___,12,16,20{\displaystyle 12,16,20}…, the term immediately following the space is 12. Subtract the common difference of 4 from this term to find 12−4=8{\displaystyle 12-4=8}. The result of 8 should fill in the blank space.

In the working example, the two results of 4+4{\displaystyle 4+4} and 12−4{\displaystyle 12-4} both gave the solution of 8. Therefore, the missing term in this arithmetic sequence is 8. The full sequence is 0,4,8,12,16,20{\displaystyle 0,4,8,12,16,20}….

It is common in working with arithmetic sequences to use the variable a(1) to designate the first term of a sequence. You may, of course, choose any variable you like, and the results should be the same. For example, given the sequence 3,8,13,18{\displaystyle 3,8,13,18}…, the first term is 3{\displaystyle 3}, which can be designated algebraically as a(1).

The term a(n) can be read as “the nth term of a,” where n represents which number in the list you want to find and a(n) is the actual value of that number. For example, if you are asked to find the 100th item in an arithmetic sequence, then n will be 100. Note that n is 100, in this example, but a(n) will be the value of the 100th term, not the number 100 itself.

For example, in the working example 3,8,13,18{\displaystyle 3,8,13,18}…, we know that a(1) is the first term 3, and the common difference d is 5. Suppose you are asked to find the 100th term in that sequence. Then n=100, and (n-1)=99. The complete explicit formula, with the data filled in, is then a(100)=3+(99)(5){\displaystyle a(100)=3+(99)(5)}. This simplifies to 498, which is the 100th term of that sequence.

For example, suppose you have the end of a list of numbers, but you need to know what the beginning of the sequence was. You can rearrange the formula to give you a(1)=a(n)−(n−1)d{\displaystyle a(1)=a(n)-(n-1)d} If you know the starting point of an arithmetic sequence and its ending point, but you need to know how many terms are in the list, you can rearrange the explicit formula to solve for n. This would be n=a(n)−a(1)d+1{\displaystyle n={\frac {a(n)-a(1)}{d}}+1}. If you need to review the basic rules of algebra to create this result, check out Learn Algebra or Simplify Algebraic Expressions.

Use the equation a(1)=(n−1)d−a(n){\displaystyle a(1)=(n-1)d-a(n)}, and fill in the information that you know. Since you know that the 50th term is 300, then n=50, n-1=49 and a(n)=300. You also are given that the common difference, d, is 7. Therefore, the formula becomes a(1)=(49)(7)−300{\displaystyle a(1)=(49)(7)-300}. This works out to 343−300=43{\displaystyle 343-300=43}. The sequence that you have began at 43, and counted up by 7. Therefore, it looks like 43,50,57,64,71,78…293,300.

Suppose you know that a given arithmetic sequence begins at 100 and increases by 13. You are also told that the final term is 2,856. To find the length of the sequence, use the terms a1=100, d=13, and a(n)=2856. Insert these terms into the formula to give n=2856−10013+1{\displaystyle n={\frac {2856-100}{13}}+1}. If you work this out, you get n=275613+1{\displaystyle n={\frac {2756}{13}}+1}, which equals 212+1, which is 213. There are 213 terms in that sequence. This sample sequence would look like 100, 113, 126, 139… 2843, 2856.