Arithmetic Sequences and Series | ||||||||||||||||||||||||||||

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Introduction | ||||||||||||||||||||||||||||

This page will teach you about arithmetic sequences and series. Here are the sections within this page: - Identifying Arithmetic Sequences
- Formulas for the Nth Term: Recursive and Explicit Rules
- Finding a Formula for an Arithmetic Sequence
- Finding the Number of Terms in an Arithmetic Sequence
- Finding the Sum of Arithmetic Series
- Instructional Videos
- Interactive Quizmasters
- Related Lessons and Quizmasters
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Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. The following sequences are arithmetic sequences: Sequence A: 5 , 8 , 11 , 14 , 17 , ... For sequence A, if we add 3 to the first number we will get the second number. This works for any pair of consecutive numbers. The second number plus 3 is the third number: 8 + 3 = 11, and so on.
For sequence B, if we add 5 to the first number we will get the second number. This also works for any pair of consecutive numbers. The third number plus 5 is the fourth number: 36 + 5 = 41, which will work throughout the entire sequence. Sequence C is a little different because we need to add -2 to the first number to get the second number. This too works for any pair of consecutive numbers. The fourth number plus -2 is the fifth number: 14 + (-2) = 12.
Because these sequences behave according to this simple rule of addiing a constant number to one term to get to another, they are called arithmetic sequences. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called the
Mathematicians also refer to generic sequences using the letter
This means that if we refer to the fifth term of a certain sequence, we will label it a
The
...where n is any positive integer greater than 1. Remember, the letter | ||||||||||||||||||||||||||||

To determine any number within an arithmetic sequence, there are two formulas that can be utilized. Here is the recursive rule.
The recursive rule means to find any number in the sequence, we must add the common difference to the previous number in this list. Let us say we were given this arithmetic sequence.
First, we would identify the common difference. We can see the common difference is 4 no matter which adjacent numbers we choose from the sequence. To find the next number after 19 we have to add 4. 19 + 4 = 23. So, 23 is the 6th number in the sequence. 23 + 4 = 27; so, 27 is the 7th number in the sequence, and so on...
What if we have to find the 724th term? This method would force us to find all the 723 terms that come before it before we could find it. That would take too long. So, there is a better formula. It is called the
So, if we want to find the 724th term, we can use this
The rule gives us a | ||||||||||||||||||||||||||||

Each arithmetic sequence has its own unique formula. The formula can be used to find any term we with to find, which makes it a valuable formula. To find these formulas, we will use the explicit rule. Let us also look at the following examples.
If we match each term with it's corresponding term number, we get:
The fixed number, called the _{n} = a_{1} + (n - 1)da _{n} = 5 + (n - 1)(3)a _{n} = 5 + 3n - 3a _{n} = 3n + 2
_{37} like this...
_{n} = 3n + 2a _{37} = 3(37) + 2a _{37} = 111 + 2a _{37} = 113
Example 2: Find a formula that defines the nth term for sequence B.
Here is sequence B.
We can identify a few facts about it. Its first term, a
_{n} = a_{1} + (n - 1)da _{n} = 26 + (n - 1)(5)a _{n} = 26 + 5n - 5a _{n} = 5n + 21
_{n} = 5n + 21a _{14} = 5(14) + 21a _{14} = 70 + 21a _{14} = 91
uizmaster: Finding Formula for General Term | ||||||||||||||||||||||||||||

It may be necessary to calculate the number of terms in a certain arithmetic sequence. To do so, we would need to know two things. We would need to know a few terms so that we could calculate the common difference and ultimately the formula for the general term. We would also need to know the last number in the sequence. Once we know the formula for the general term of a sequence and the last term, the procedure involves the use of algebra. Use the two examples below to see how it is done.
This is sequence A. In the previous section, we found the formula to be a _{n} = 3n + 247 = 3n + 2 45 = 3n 15 = n n = 15
Our first task is to find the formula for this sequence given a _{n} = a_{1} + (n - 1)da _{n} = 20 + (n - 1)(-2)a _{n} = 20 - 2n + 2a _{n} = -2n + 22
_{n}, is -26. Substituting this into the formula gives us...
_{n} = -2n + 22-26 = -2n + 22 -48 = -2n 24 = n n = 24
ideo: Finding the Number of Terms in a Finite Arithmetic Sequence | ||||||||||||||||||||||||||||

Given our generic arithmetic sequence...
...we can add the terms, called a
There exists a formula that can add such a finite list of these numbers. It requires three pieces of information. The formula is...
...where S If you would like to see a derivation of this arithmetic series sum formula, watch this video.
ideo: Arithmetic Series: Deriving the Sum Formula Usually problems present themselves in either of two ways. Either the first number and the last number of the sequence are known or the first number in the sequence and the number of terms are known. The following two problems will explain how to find a sum of a finite series.
In order to use the sum formula. We need to know a few things. We need to know n, the number of terms in the series. We need to know a
To find the n-value, let's use the formula for the series. We already determined the formula for the sequence in a previous section. We found it to be a _{n} = 3n + 2128 = 3n + 2 126 = 3n 42 = n n = 42 _{1} = 5, and a_{42} = 128. We can substitute these number into the sum formula, like so.
_{n} = (1/2)n(a_{1} + a_{n})S _{42} = (1/2)(42)(5 + 128)S _{42} = (21)(133)S _{42} = 2793
First we have to figure out what our series looks like. We need to write multiples of seven and add them together, like this.
To find the last number in the series, which we need for the sum formula, we have to develop a formula for the series. So, we will use the _{n} = a_{1} + (n - 1)da _{n} = 7 + (n - 1)(7)a _{n} = 7 + 7n - 7a _{n} = 7n_{n} = 7na _{n} = 7(205)a _{n} = 1435
To find the sum, we will substitute information into the sum formula. We will substitute a _{n} = (1/2)n(a_{1} + a_{n})S _{42} = (1/2)(205)(7 + 1435)S _{42} = (1/2)(205)(1442)S _{42} = (1/2)(1442)(205)S _{42} = (721)(205)S _{42} = 147805
ideo: Arithmetic Sequence: Finding the Sum |

Watch these instructional videos.
ideo: Finding the nth Term of an Arithmetic Sequence |

After reading the lesson, try our quizmaster. MATHguide has developed numerous testing and checking programs to solidify skills demonstrated in this lesson. The following quizmasters are available:
uizmaster: Finding Formula for General Term |

After reading the lesson, try a related lesson or quizmaster.
esson: Geometric Sequences and Series
uiz: Determine the Function |