Different Types of Sequences
- Let’s look at other types of sequences.
3.1: Remembering Function Notation
Consider the function \(f\) given by \(f(n) = 3n - 7\). This function takes an input, multiplies it by 3, then subtracts 7.
- \(f(10) - 1\)
- \(f(10 - 1)\)
- \(f(5) - f(4)\)
3.2: Three Sequences
Here are the values of the first 5 terms of 3 sequences:
- \(A\): 30, 40, 50, 60, 70, . . .
- \(B\): 0, 5, 15, 30, 50, . . .
- \(C\): 1, 2, 4, 8, 16, . . .
- For each sequence, describe a way to produce a new term from the previous term.
- If the patterns you described continue, which sequence has the second greatest value for the 10th term?
- Which of these could be geometric sequences? Explain how you know.
Elena says that it’s not possible to have a sequence of numbers that is both arithmetic and geometric. Do you agree with Elena? Explain your reasoning.
3.3: Representing a Sequence
Jada and Mai are trying to decide what type of sequence this could be:
Jada says: “I think this sequence is geometric because in the value column each row is 3 times the previous row.”
Mai says: “I don’t think it is geometric. I graphed it and it doesn’t look geometric.”
Do you agree with Jada or Mai? Explain or show your reasoning.
Consider the sequence 2, 5, 8, . . . How would you describe how to calculate the next term from the previous?
In this case, each term in this sequence is 3 more than the term before it.
This is an example of an arithmetic sequence. An arithmetic sequence is one where the value of each term is the value of the previous term with a constant added. If you know the constant to add, you can use it to find other terms.
For example, each term in this sequence is 3 more than the term before it. To find this constant, sometimes called the rate of change or common difference, you can subtract consecutive terms. This can also help you decide whether a sequence is arithmetic.
For example, the sequence 3, 6, 12, 24 is not an arithmetic sequence because \(6-3 \neq 12-6 \neq 24-12\). But the sequence 100, 80, 60, 40 is because if the differences of consecutive terms are all the same: \(80-100 = 60-80 = 40-60 = \text-20\). This means that the rate of change is -20 for the sequence 100, 80, 60, 40.
It is important to remember that while the last two lessons have introduced geometric and arithmetic sequences, there are many other sequences that are neither geometric nor arithmetic.
- arithmetic sequence
A sequence in which each term is the previous term plus a constant.
- geometric sequence
A sequence in which each term is a constant times the previous term.