13.2 Arithmetic sequences  (Page 4/8)

Solving application problems with arithmetic sequences

In many application problems, it often makes sense to use an initial term of $a_0$ instead of $a_1.$ In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

Key equations

Key concepts

• An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
• The constant between two consecutive terms is called the common difference.
• The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See [link] .
• The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See [link] and [link] .
• A recursive formula for an arithmetic sequence with common difference $d$ is given by $a_n=a_{n-1}+d,n\ge 2.$ See [link] .
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for an arithmetic sequence with common difference $d$ is given by $a_n=a_1+d(n-1).$ See [link] .
• An explicit formula can be used to find the number of terms in a sequence. See [link] .
• In application problems, we sometimes alter the explicit formula slightly to $a_n=a_0+dn.$ See [link] .

Section exercises

Verbal