8.1. Sequences http://www.ck12.org
The terms of the sequence are:
n 1 2 3 4 ...
an= n2
n+ 112
1 + 1 =
1
2
22
2 + 1 =
4
3
9
4
16
5 ...
You can also find the rule for a sequence.
Example 3
Find the rule for the sequence below.
n 1 2 3 4 ...
an=?^12 −^2334 −^45 ...Look at each term in terms of its index. The numerator of each term matches the index. The denominator is one
more than the index. So far, we can write the formulaanasn+n 1. However, we are not done. Notice that each
even-indexed term has a negative sign. This means that all of terms of the sequence have a power of−1. The powers
of−1 alternate between odd and even. Usually, alternating powers of−1 can be denote by(− 1 )nor(− 1 )n+^1. Since
the terms are negative for even indices, we use(− 1 )n+^1. Thus, the rule for the sequence isan=(−n^1 +)n 1 +^1 n. You can
check the rule by finding the first few terms of the sequencean=(−n^1 +)n+ 11 n.
Limit of a Sequence
We are interested in the behavior of the sequence as the value ofngets very large. Many times a sequence will get
closer to a certain number, orlimit,asngets large. Finding the limit of a sequence is very similar to finding the limit
of a function. Let’s look at some graphs of sequences.
Example 4
Find the limit of the sequence{ 2 n^1 + 1 }asngoes to infinity.
Solution
We can graph the corresponding functiony= 2 n^1 + 1 forn= 1 , 2 , 3 ,....The graph of is similar to the continuous
functiony= 2 x^1 + 1 for the domain ofx≥1.