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Arithmetic Sequences and Series

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

  • about arithmetic sequences
  • how to find the formula for the nth term of an arithmetic sequence
  • about arithmetic series
  • how to find the sum of an arithmetic series

What is an Arithmetic Sequence?
A list of numbers that follows a rule is called a sequence. Sequences whose rule is the addition of a constant are called arithmetic sequences, similar to geometric sequences that follow a rule of multiplication.
Homework problems on arithmetic sequences often ask us to find the nth term of a sequence using a formula. Arithmetic sequences are important to understanding arithmetic series.

Arithmetic Sequences
Determine the nth term of an arithmetic sequence.
Determine the common difference of an arithmetic sequence.
Determine the formula for an arithmetic sequence.
An arithmetic sequence is a sequence that has the pattern of adding a constant to determine consecutive terms. We say arithmetic sequences have a common difference.
Examples:
1. A sequence is a function. What is the domain and range of the following sequence?

2. Given the formula for the arithmetic sequence, determine the first 3 terms and the 8th term. Also state the common difference.
an = -4n + 3

3. Given the arithmetic sequence, determine the formula and the 12th term
-2,1.5,5,8.5,12,15.5, ...
Introduction to Arithmetic Sequences
Just a quick idea of what an arithmetic sequence is and a few examples
Examples:
Determine which of the following sequences are arithmetic. If they are arithmetic, give the value of 'd'. 1) 3,8,13,18,23,28,33, ...
2) -.7, -1.7, -2.7, -3.7, -4.7, ...
3) 1.6, 2.2, 2.8, 3.3, 3.9, 4.5, ...
4) 4/3, 5/3, 2, 7/3, 8/3, 3, ...



Arithmetic Sequences: A Formula for the 'nth' Term
This video derives the formula to find the 'n-th' term of a sequence by considering an example.
The formula is then used to do a few different problems.
Example:
Suppose we have the arithmetic sequence
3, 8, 13, 18, 23, 28, 33, ...
Find
a10
a202
How to find the general term of an arithmetic sequence?
an = a1 + (n-1)d


What is an Arithmetic Series?
We can use what we know of arithmetic sequences to understand arithmetic series. An arithmetic series is a series or summation that sums the terms of an arithmetic sequence.
There are methods and formulas we can use to find the value of an arithmetic series.
Understanding arithmetic series can help to understand geometric series, and both concepts will be used when learning more complex Calculus topics.
Arithmetic Series
Define a series.
Determine the partial sum of an arithmetic series.
Summing or adding the terms of an arithmetic sequence creates what is called a series.
Sn = n/2(an + an)
Examples:
Determine the sum of the arithmetic series
1. 3 + 8 + 13 + ... + 73
2. an = -4n + 3; n = 20
How to find the sum of an arithmetic series?
How to find the sum of an arithmetic series when you're given only the first few terms and the last one?
Example:
Find the sum
-82, -80, -78, ... +64, +66


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