Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

4.18.3 Define an Arithmetic Sequence by the n^th Term

4.18.3 • Define an Arithmetic Sequence by the nth Term

Activity

1. Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again . . .

a. Instead of writing a recursive definition, Clare writes $C (n) = 80 \cdot (\frac{1}{2})^{n}$ , where $C$ is the area, in square inches, of the paper after $n$ cuts. Explain where the different terms in her expression came from.

Compare your answers:

Each cut halves the area, and there are $n$ cuts. The original area is 80, so $80 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \cdot \cdot \cdot \cdot \frac{1}{2}$ with $n$ multiplications of $\frac{1}{2}$ is the same as saying $80 \cdot (\frac{1}{2})^{n}$ .

b. Approximately what is the area of the paper after 10 cuts?

Compare your answers:

Approximately 0.078 square inches (exactly $\frac{80}{1024}$ inches)

2. Kiran takes a piece of paper with length 8 inches and width 10 inches and cuts away 1 inch of the width. Then he does it again, and again . . .

a. Complete the table for the area of Kiran’s paper $K (n)$ , in square inches, after $n$ cuts.

$n$	$K (n)$
0	80
1
2	$80 - 8 - 8 = 80 - 8 (2) = 64$
3
4
5

Compare your answers:

When $n$ is 1, $K (n)$ is 72. When $n$ is 3, $K (n)$ is 56. When $n$ is 4, $K (n)$ is 48. When $n$ is 5, $K (n)$ is 40.

b. Kiran says the area after 6 cuts, in square inches, is $80 - 8 \cdot 6$ . Explain where the different terms in his expression came from.

Compare your answers:

Each cut removes 8 square inches, so 6 cuts removes 48 square inches. The original area was 80 square inches, so the remaining area, in square inches, is $80 - 8 \cdot 6$ .

c. Write a definition for $K (n)$ that is not recursive.

Compare your answers:

$K (n) = 80 - 8 n$ for integers $0 \leq n \leq 9$ or equivalent.

3. Which is larger, $K (6)$ or $C (6)$ ?

Compare your answer:

$K (6)$ is larger.

4. Is $K (n)$ arithmetic or geometric?

Compare your answer:

$K (n)$ is arithmetic.

5. Is $C (n)$ arithmetic or geometric?

Compare your answer:

$C (n)$ is geometric.

A recursive sequence is a sequence in which terms are defined using one or more of the previous terms. If you know the $n^{th}$ term of an arithmetic or geometric sequence and the common difference or factor, you can find the ${(n + 1)}^{th}$ term by using the recursive formula.

Different definitions can often create the same sequence. For arithmetic and geometric sequences, there are general rules, called explicit rules, that can be followed to help find any term in the sequences.

These are called the nth term or the general term of the sequence.

Arithmetic Sequence Formulas

Recursive Formula

$a_{n} = a_{n - 1} + d$ , $a_{1}$ = first term

where $a_{1}$ is the first term, $n$ is the term you want, and $d$ is the common difference.

Explicit General Formula

$a_{n} = a_{1} + (n - 1) d$

Where $a_{1}$ is the first term, $n$ is the term you want, and $d$ is the common difference.

Self Check

What is the 12th term in an arithmetic sequence where the first term is 22 and the common difference is –2?

$-45056$
$0$
$-22$
$-2$

Additional Resources

The

n^{th}

Term of Arithmetic Sequences

GENERAL TERM ( $n^{th}$ TERM) OF AN ARITHMETIC SEQUENCE

The general term of an arithmetic sequence with first term $a_{1}$ and common difference $d$ is: $a_{n} = a_{1} + (n - 1) d$ .

Example 1

Write the explicit formula for the sequence given by the terms 18, 21, 24, 27, ...

Solution

Step 1- Write the general formula.

$a_{n} = a_{1} + (n - 1) d$

Step 2 - Substitute values for the first term, common difference/ratio, term number.

$a_{1} = 18$ , $d = 3$

$a_{n} = 18 + (n - 1) 3$

So, the explicit formula for the sequence 18, 21, 24, 27… is $a_{n} = 18 + (n - 1) 3$ .

Now any term can more easily be determined - even the $1000^{th}$ term!

Example 2

Find the $15^{th}$ term of a sequence where the first term is 3 and the common difference is 6.

Step 1 - Write the general formula.

$a_{n} = a_{1} + (n - 1) d$

Step 2 - Substitute values for the first term, common difference/ratio, term number.

$a_{1} = 3, d = 6$ , $n = 15$

$a_{15} = 3 + (15 - 1) 6$

Step 3 - Simplify the expression.

$a_{15} = 3 + (14) 6$

$a_{15} = 3 + 84$

$a_{15} = 87$

If the explicit formula for this question was needed, we would not have substituted $n = 15$ and the $n$ th term formula would have been $a_{n} = 3 + (n - 1) 6$ . This formula is also equivalent to

$a_{n} = 3 + 6 (n - 1)$

Try it

Try It: The $n^{th}$ Term of Arithmetic Sequences

Find the formula for the $n$ th term of a sequence where the first term is 7 and the common difference is 9.

Compare your answer:

Here is how to find the $27^{th}$ term of the sequence:

Step 1 - Write the general formula.

$a_{n} = a_{1} + (n - 1) d$

Step 2 - Substitute values for the first term, common difference/ratio, term number.

$a_{1} = 7$ , $d = 9$

$a_{n} = 7 + (n - 1) 9$

4.18.3 Define an Arithmetic Sequence by the nth Term