Find Any Term of an Arithmetic Sequence

<math>Insert formula here</math>An arithmetic sequence is any list of numbers that differ, from one to the next, by a constant amount. For example, the list of even numbers, <math>0,2,4,6,8</math>… is an arithmetic sequence, because the difference from one number in the list to the next is always 2. If you know you are working with an arithmetic sequence, you may be asked to find the very next term from a given list. You may also be asked to fill in a gap where a term is missing. Finally, you might want to know, for example, the 100th term, without actually writing out all 100 terms. A few simple steps can help you do any of these.

Steps

Finding the Next Term in an Arithmetic Sequence

  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.[1]
    • For example, suppose you have the list <math>1,4,7,10,13</math>.... Subtract <math>4-1</math> to find the common difference of 3.
    • Suppose you have a list of terms that decreases, such as <math>25,21,17,13</math>…. You still subtract the first term from the second to find the difference. In this case, that gives you <math>21-25= -4</math>. 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. 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, <math>1,4,7,10,13</math>… choose the second and third terms of the list. Subtract <math>7-4</math>, and you find that the difference is still 3. To confirm, check one more example and subtract <math>13-10</math>, 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 <math>1,2,3,6,9</math>…. 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.
  3. Add the common difference to the last given term. 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 <math>1,4,7,10,13</math>…, to find the next number in the list, add the common difference of 3 to the last given term. Adding <math>13+3</math> 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 <math>1,4,7,10,13,16,19,22,25</math>…. You can do this as long as you like.

Finding a Missing Internal Term

  1. Verify that you are starting with an arithmetic sequence. In some cases, you may have a list of numbers with a missing term in the middle. Begin, as before, by checking that your list is an arithmetic sequence. Select any two consecutive terms and find the difference between them. Then check this against two other consecutive terms in the list. If the differences are the same, you can presume that you are working with an arithmetic sequence and proceed.
    • For example, suppose you have the list <math>0,4</math>,___,<math>12,16,20</math>…. Start by subtracting <math>4-0</math> to find a difference of 4. Check this against two other consecutive terms, such as <math>16-12</math>. The difference is again 4. You can proceed.
  2. Add the common difference to the term before the space. This is similar to adding a term to the end of a sequence. Find the term that immediately precedes the space in your sequence. This is the “last” number that you know. Add your common difference to this term, to find the number that should fill in the space.
    • In our working example, <math>0,4</math>,____,<math>12,16,20</math>…, the term preceding the space is 4, and our common difference for this list is also 4. So add <math>4+4</math> to get 8, which should be the number in the blank space.
  3. Subtract the common difference from the term following the space. To be sure that you have the correct answer, check from the other direction. An arithmetic sequence should be consistent going in either direction. If you move from left to right and add 4, then going in the opposite direction, from right to left, you would do the opposite and subtract 4.
    • In the working example, <math>0,4</math>,___,<math>12,16,20</math>…, the term immediately following the space is 12. Subtract the common difference of 4 from this term to find <math>12-4 = 8</math>. The result of 8 should fill in the blank space.
  4. Compare your results. The two results that you get, from adding up from the bottom or from subtracting down from the top should match. If they do, then you have found the value for the missing term. If they do not, then you need to check your work. You may not have a true arithmetic sequence.
    • In the working example, the two results of <math>4+4</math> and <math>12-4</math> both gave the solution of 8. Therefore, the missing term in this arithmetic sequence is 8. The full sequence is <math>0,4,8,12,16,20</math>….

Finding the Nth Term of an Arithmetic Sequence

  1. Identify the first term of the sequence. Not every sequence begins with the numbers 0 or 1. Look at the list of numbers that you have and find the first term. This is your starting point, which can be designated using variables as a(1).
    • 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 <math>3,8,13,18</math>…, the first term is <math>3</math>, which can be designated algebraically as a(1).
  2. Define your common difference as d. Find the common difference for the sequence as before. In this working example, the common difference is <math>8-3</math>, which is 5. Checking with other terms in the sequence provides the same result. We will note this common difference with the algebraic variable d.
  3. Use the explicit formula. An explicit formula is an algebraic equation that you can use to find any term of an arithmetic sequence, without having to write out the full list. The explicit formula for an algebraic sequence is <math>a(n)=a(1)+(n-1)d</math>.
    • 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.
  4. Fill in your information to solve the problem. Using the explicit formula for your sequence, fill in the information that you know to find the term that you need.
    • For example, in the working example <math>3,8,13,18</math>…, 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 <math>a(100)=3+(99)(5)</math>. This simplifies to 498, which is the 100th term of that sequence.

Using the Explicit Formula to Find Additional Information

  1. Rearrange the explicit formula to solve for other variables. Using the explicit formula and some basic algebra, you can find several pieces of information about an arithmetic sequence. In its original form, <math>a(n)=a(1)+(n-1)d</math>, the explicit formula is designed to solve for an and give you the nth term of a sequence. However, you can algebraically manipulate this formula and solve for any of the variables.
    • 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 <math>a(1)=(n-1)d-a(n).</math>
    • 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 <math>n=\frac{a(n)-a(1)}{d}+1</math>.
    • If you need to review the basic rules of algebra to create this result, check out Do Algebra or Simplify Algebraic Expressions.
  2. Find the first term of a sequence. You may know that the 50th term of an arithmetic sequence is 300, and you know that the terms have been increasing by 7 (the “common difference”), but you want to find out what the first term of the sequence was. Use the revised explicit formula that solves for a1 to find your answer.
    • Use the equation <math>a(1)=(n-1)d-a(n)</math>, 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 <math>a(1)=(49)(7)-300</math>. This works out to <math>343-300=43</math>. 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.
  3. Find the length of a sequence. Suppose you know all about the start and end of an arithmetic sequence, but you need to find out how long it is. Use the revised formula <math>n=\frac{a(n)-a(1)}{d}+1</math>.
    • 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 <math>n=\frac{2856-100}{13}+1</math>. If you work this out, you get <math>n=\frac{2756}{13}+1</math>, 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.

Warnings

  • There are different kinds of sequences of numbers. Do not assume that a list of numbers is an arithmetic sequence. Always check at least two pairs of terms, or preferably three or four, to find the common difference between terms.


Tips

  • Remember that d can be either positive or negative, depending if it is being added or subtracted.

Related Articles

Sources and Citations

  1. ↑ http://www.algebralab.org/lessons/lesson.aspx?file=algebra_arithseq.xml
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