Different Types of Sequences

Different Types of Sequences

By Zinaida Avdic

Multiple Choice

Consider the function f given by f(n)= 3n -7. This function takes an input n, multiplies it by 3, then subtracts 7 to get an output .

Evaluate f(5) -f(4).

• 23

• 22

• 20

• 3

• ​I can explain what it means for a sequence to be arithmetic or geometric.

​Learning Target:

Open Ended

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, . . . 1.

For each sequence, describe a way to produce a new term from the previous term.

​Sample response:
For A , each term is 10 more than the previous term.
For B , to get a term, add 1 to the previous term number, multiply by 5, and add that number to the previous term (add 5, then add 10, then add 15, etc.).
For C, each term is double the previous term.

Multiple Choice

Which of these could be geometric sequences?

• A

• B

• C

• Neither

​ A geometric sequence has the same multiplier from one term to the next (the growth factor), and neither A nor B has this. Sequence C does. The growth factor is 2.

​• C is geometric but A is not.
• In A, we always add 10 to get from term to term, but in C, we always multiply by 2.
• In C , the growth factor is 2. In A, we get the next term by adding 10 to the previous term.

Sequence A is an example of an arithmetic sequence. Here are two ways to know a sequence is arithmetic:
• We always add the same number to get from one term to the next.
• If we subtract any term from the next term, we always get the same number.

Multiple Choice

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?

• I agree with Jada

• I agree with Mai

• Neither Jada nor Mai are correct

Jada noticed that each value was multiplied by 3 to get to the next row, but the table skips terms. If the sequence were geometric, it would have to be 2, 6, 18, 54, 108. When Mai made a graph, she could see the gap in the sequence.

​Without knowing the missing terms, we don’t know for sure if the sequence is arithmetic, but it could be since the three given points are all on the same line with slope 4. The missing points could be(3,10) and (4,14), so that the sequence would be 2, 6, 10, 14, 18.

Open Ended

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.

Lesson Synthesis:

-Both arithmetic and geometric sequences are types of sequences. That is, they are both lists of numbers.
- They both have a starting term.
- For both arithmetic and geometric sequences, to get from one term to the next, we “do the same thing each time.”
- For geometric sequences, we always multiply by the same number to get the next term. For arithmetic sequences, we always add the same number to get the next term.

- Geometric sequences have growth factors (the quotient of any term and the previous term), while arithmetic sequences have a constant rate of change (the difference of any term and the previous term).

Multiple Choice

Sequence 200, 40, 5 is:

• geometric

• arithmetic

• neither

Multiple Choice

Sequence 2,4, 16 is:

• geometric

• neither

• arithmetic

Multiple Choice

Sequence 10, 20, 30 is:

• geometric

• arithmetic

• neither

Fill in the Blank

Sequence 100, 20, 4 is ___________________.

Open Ended

What are the next three terms of the sequence -2,4....

Fill in the Blank

The next three terms of the sequence 5, -4 are ___________________

-

,

-

,

-

.