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

Activity

Converting from a Recursive Formula to an Explicit Formula for Arithmetic Sequences

An arithmetic sequence has the following recursive formula:

$a_{1} = 3$

$a_{n} = a_{n - 1} + 2$ , $n \geq 2$

Recall that this formula gives us the following two pieces of information:

The first term is 3.
To get any term from its previous term, add 2. In other words, the common difference is 2.

Let’s find an explicit formula for the sequence.

We can represent a sequence whose first term is $A$ and common difference is $B$ with the standard explicit form $A + B (n - 1)$ .

Therefore, an explicit formula of the sequence is $a_{n} = 3 + 2 (n - 1)$ .

1. Write an explicit (nth term) formula for an arithmetic sequence where $b_{1} = 2$ and $b_{n} = b_{n - 1} + 7$ , $n \geq 2$ .

Compare your answer:

$b_{n} = 2 + (n - 1) 7$

Converting from a Explicit Formula to an Recursive Formula for Arithmetic Sequences

Now try to take the nth term formula and use it to write the recursive formula of an arithmetic sequence.

We are given the following nth term of an arithmetic sequence:

$d_{n} = 5 + (n - 1) \cdot 16$

This formula is given in the standard explicit form $A + B (n - 1)$ , where $A$ is the first term and $B$ is the common difference. Therefore, the first term of the sequence is 5, and the common difference is 16.

Let’s find a recursive formula for the sequence. Recall that the recursive formula gives us two pieces of information:

The first term, which we know is 5.
The pattern rule to get any term from the term that comes before it, which we know is “add 16.”

Therefore, this is a recursive formula for the sequence: $d_{1} = 5$ , $d_{n} = d_{n - 1} + 16$ , $n \geq 2$ .

2. Write the recursive formula of $e_{n} = 12 + (n - 1) 2$ .

Compare your answer:

$e_{1} = 12$ , $e_{n} = e_{n - 1} + 2$

$n \geq 2$

Converting from a Recursive Formula to an Explicit Formula for Geometric Sequences

Now change the recursive formula of a geometric sequence into the nth term.

Here is the recursive formula of a geometric sequence, $g (n)$ :

$g_{1} = 9$ , $g_{n} = g_{n - 1} \cdot (8)$ , $n \geq 2$

9 is the first term of the sequence, and 8 is the common ratio.

An explicit formula is structured as: $g (x) =$ (1st term of sequence) $\cdot$ (common ratio) $^{(x - 1)}$ .

Substitute the values, and you get the explicit formula: $g_{n} = 9 \cdot 8^{(n - 1)}$ .

3. Write the nth term of the geometric sequence where $g_{1} = 5$ and $g_{n} = g_{n - 1} \cdot 4$ , $n \geq 2$ .

Compare your answer:

$g_{n} = 5 \cdot 4^{(n - 1)}$

Converting from an Explicit Formula to a Recursive Formula for Geometric Sequences

Finally, geometric sequences represented by the nth term can be represented with the recursive definition.

Start with the nth term formula for $g_{n}$ where $g_{n} = 6 \cdot 3^{n - 1}$ .

Working backward, when $n = 1$ , $g_{1} = 6$ . This is the initial term.

The formula for geometric sequences recursively is $g_{n} = g_{n - 1} \cdot r$ , where $r$ is the common ratio.

The common ratio of the given sequence is 3.

So, $g_{1} = 6$ and $g_{n} = g_{n - 1} \cdot 3$ , $n \geq 2$ .

4. Write the geometric sequence represented by $g_{n} = 14 \cdot 2^{n - 1}$ using the recursive definition.

Compare your answer:

$g_{1} = 14$ , $g_{n} = g_{n - 1} \cdot 2$

$n \geq 2$

Why Should I Care?

A hand holds a tablet displaying a split image: on the left, wind turbines on a grassy hill; on the right, rows of solar panels under a clear sky, all against an orange background.

One of the tools to fight climate change is renewable energy. Wind farms and solar power can produce can produce clean energy. But what happens when there is no wind or sunshine? Energy companies rely on battery storage systems to save energy that is created on sunny or windy days, so it can be used when the sun and wind aren't cooperating.

It is important that the batteries can store as much energy as possible, for as long as possible. Scientists who create these batteries use functions to help them calculate energy inputs and outputs and the changes that make a battery lose power.

Self Check

Write the nth term formula of the geometric sequence represented by $g_1=-3$ and $g_n=g_{n-1} \cdot 6$ , $n\geq2$ .

$g_n=6(n-1)-3$
$g_n=-3 \cdot 6^{n-1}$
$g_n=-3 \cdot 6^n$
$g_n=6n+-3$

Additional Resources

Switching Between Definitions of Sequences

A recursive formula defines each term of a sequence using the previous terms.

Sequence	Recursive Formula	Explicit or nth Term Formula
Arithmetic	$F_{1} = a_{1}$ , $F_{n} = F_{n - 1} + d$ , where $a_{1}$ is the first term and $d$ is the common difference.	$a_{n} = a_{1} + (n - 1) d$ , where $a_{1}$ is the first term, $n$ is the term you want, and $d$ is the common difference.
Geometric	$H_{1} = a_{1}$ , $H_{n} = H_{n - 1} \cdot r$ , Where $a_{1}$ is the first term and $r$ is the common ratio.	$a_{n} = a_{1} r^{n - 1}$ , Where $a_{1}$ is the first term, $n$ is the term you want, and $r$ is the common ratio.

Example 1

Find the explicit general formula for the arithmetic sequence given in recursive form.

$a_{1} = 5$ , $a_{n} = a_{n - 1} + 6$ , $n \geq 2$

Step 1 - Write the recursive formula. (given)

$a_{1} = 5$ , $a_{n} = a_{n - 1} + 6$ , $n \geq 2$

Step 2 - Identify the first term.

$a_{1} = 5$

Step 3 - Identify the common difference/ratio.

$d = 6$

Step 4 - Write the explicit formula for the $n^{t h}$ term.

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

Step 5 - Substitute the first term and common difference/ratio into the formula.

$a_{n} = 5 + (n - 1) 6$

Check by finding the first 3 terms using each formula:

$a_{1} = 5$ , $a (n) = a_{n} = a_{n - 1} + 6$ , $n \geq 2$	$a_{1} = 5$	$a_{2} = 5 + 6 = 11$	$a_{3} = 11 + 6 = 17$
$a_{n} = 5 + (n - 1) 6$	$a_{1} = 5$	$a_{2} = 5 + (2 - 1) 6 = 11$	$a_{3} = 5 + (3 - 1) 6 = 17$

Both formulas produce the same first 3 terms: 5, 11, 17.

Example 2

Find the explicit general formula for the geometric sequence given in recursive form.

$b_{1} = 4$ , $b_{n} = b_{n - 1} \cdot 5$ , $n \geq 2$

Step 1 - Write the recursive formula (given).

$b_{1} = 4$ , $b_{n} = b_{n - 1} \cdot 5$ , $n \geq 2$

Step 2 - Identify the first term.

$a_{1} = 4$

Step 3 - Identify the common difference/ratio.

$r = 5$

Step 4 - Write the explicit formula for the $n^{t h}$ term.

$a_{n} = a_{1} r^{n - 1}$

Step 5 - Substitute the first term and the common difference/ratio into the formula.

$a_{n} = 4 (5)^{n - 1}$

Check by finding the first 3 terms using each formula:

$b_{1} = 4$ ,

$b_{n} = b_{n - 1} \cdot 5$ ,

$n \geq 2$

b_{1} = 4

$b_{2} = b_{2 - 1} \cdot 5 =$

$4 \cdot 5 = 20$

$b_{3} = b_{3 - 1} \cdot 5 =$

$20 \cdot 5 = 100$

$a_{n} = 4 (5)^{n - 1}$

a_{1} = 4 (5)^{1 - 1} = 4

a_{2} = 4 (5)^{2 - 1} = 20

a_{3} = 4 (5)^{3 - 1} = 100

Both formulas produce the same first 3 terms: 4, 20, 100.

Example 3

Find the recursive formula for the arithmetic sequence given in explicit general form.

$a_{n} = 21 + (n - 1) (- 2)$

Step 1 - Write the explicit formula (given).

$a_{n} = 21 + (n - 1) (- 2)$

Step 2 - Identify the first term.

$a_{1} = 21$

Step 3 - Identify the common difference/ratio.

$d = - 2$

Step 4 - Write the explicit formula for the $n^{t h}$ term.

$F_{1} = a_{1}$ , $F_{n} = F_{n - 1} + d$ , $n \geq 2$

Step 5 - Substitute the first term and the common difference/ratio into the formula.

$F_{1} = 21$ , $F_{n} = F_{n - 1} - 2$ , $n \geq 2$

Check your reasoning by finding the first three terms using each formula. Both formulas produce the same first 3 terms: 21, 19, 17.

Example 4

Find the recursive formula for the arithmetic sequence given in explicit general form.

$a_{n} = 3 \cdot (\frac{1}{4})^{n - 1}$

Step 1 - Write the explicit formula.

$a_{n} = 3 \cdot (\frac{1}{4})^{n - 1}$

Step 2 - Identify the first term.

$a_{1} = 3$

Step 3 - Identify the common difference/ratio.

$r = \frac{1}{4}$

Step 4 - Write the recursive formula.

$H_{1} = a_{1}$ , $H_{n} = H_{n - 1} \cdot r$ , $n \geq 2$

Step 5 - Substitute the first term and the common difference/ratio into the formula.

$H_{1} = 3$ , $H_{n} = H_{n - 1} \cdot \frac{1}{4}$ , $n \geq 2$

Check your reasoning by finding the first three terms using each formula. Both formulas produce the same first 3 terms: 3, $\frac{3}{4}$ , $\frac{3}{16}$ .

Try it

Try It: Switching Between Definitions of Sequences

Write the recursive formula for the sequence below that is given in explicit form:

$a_{n} = 14 + (n - 1) (3)$

Here is how to write the sequence using the recursive definition:

Step 1 - Write the explicit formula.

$a_{n} = 14 + (n - 1) (3)$

Step 2 - Identify the first term.

$a_{1} = 14$

Step 3 - Identify the common difference.

$d = 3$

Step 4 - Write the recursive formula.

$F 1 = a_{1}$ , $F_{n} = F_{n - 1} + d$ , $n \geq 2$

Step 5 - Substitute $a_{1}$ and $d$ into the formula.

$F_{1} = 14$ , $F_{n} = F_{n - 1} + 2$ , $n \geq 2$

Both formulas give the same first 3 terms: 14, 17, 20.

4.18.5 Interchanging between Formulas

Activity

Self Check

Additional Resources

Switching Between Definitions of Sequences

Try It: Switching Between Definitions of Sequences