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Linear equations Substitution Arithmetic sequence Geometric sequence formula Recursive formulaHere you will learn about explicit formulas, including what they are, how to use them and how to create them for arithmetic and geometric sequences.

Students will first learn about explicit formulas as part of algebra in high school.

**The explicit formula **of a sequence is a formula that enables you to find any term of a sequence.

Below are a few examples of different types of sequences and their n th term formula.

**Step by step guide:** Quadratic sequences

**Step by step guide: **Cubic graph

On this page, you will look specifically at finding the n th term for an arithmetic or geometric sequence.

An arithmetic sequence (arithmetic progression) is an ordered set of numbers that have a common difference between each consecutive term. The term-to-term rule tells how you get from one term to the next. In an arithmetic sequence, the rule will always be adding or subtracting a certain number.

**Explicit formula**

- The \textbf{n}
**th term**of an**arithmetic****sequence**is:

Where,

a_{n} is the nth term (general term)

a_{1} is the first term

n is the term position

d is the common difference

For example,

6, 2, -2, -6, -10, ... a_{1}=6d = –4 (Each term is 4 less than the one before it.)

So the explicit formula is a_n=6+[-4(n-1)].

A geometric sequence (geometric progression) is an ordered set of numbers where all terms but the first share a common ratio. The term-to-term rule explains how you get from one term to the next. In an arithmetic sequence the rule will always be multiplying by a certain number.

**Explicit formula**

- The \textbf{n}
**th term**of a**geometric sequence**is:

Where,

a_{n} is the nth term (general term)

a_{1} is the first term

n is the term position

n-1 is the previous term and is an exponent

r is the common ratio

For example,

1, 2, 4, 8, 16, 32, … a_1=1 r=2So the explicit formula is a_n=1(2)^{n-1} or a_n=2^{n-1}.

How does this relate to high school math?

**Functions – Building Functions (HSF.BF.A.1)**Write a function that describes a relationship between two quantities.

**Functions – Building Functions (HSF.BF.A.2)**

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

**Functions – Linear, Quadratic and Exponential Models (HSF.LE.A.2)**

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Use this quiz to check your grade 4 to 5 students’ understanding of number patterns. 10+ questions with answers covering a range of 4th and 5th grade number patterns topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 4 to 5 students’ understanding of number patterns. 10+ questions with answers covering a range of 4th and 5th grade number patterns topics to identify areas of strength and support!

DOWNLOAD FREEIn order to generate a sequence using an explicit formula:

**Find an explicit formula.****Substitute the given value into the formula to calculate the new value.****Repeat step \bf{2} until the desired number of terms has been generated.**

A sequence is defined by the explicit formula a_n=a_1+2(n-1) and a_1=3.

Find the next four terms of the sequence.

**Find an explicit formula.**

The explicit formula is given in the question, a_n=a_1+2(n-1) and a_1=3.

This creates the explicit formula a_n=3+2(n-1).

2**Substitute the given value into the formula to calculate the new value.**

3**Repeat step \bf{2} until the desired number of terms has been generated.**

This first term is 3. The next four terms of the sequence are 5, 7, 9, 11.

A sequence is defined by the explicit formula a_n=a_1(r)^{n-1} and a_1=1 and r=4.

Find the next four terms of the sequence.

**Find an explicit formula.**

The explicit formula is given in the question, a_n=a_1(r)^{n-1} and a_1=1 and r=4.

This creates the explicit formula a_n=1(4)^{n-1} or a_n=4^{n-1}.

**Substitute the given value into the formula to calculate the new value.**

a_2=4^{2-1}=4^1=4

**Repeat step \bf{2} until the desired number of terms has been generated.**

\begin{aligned}& a_3=4^{3-1}=4^2=16 \\\\ & a_4=4^{4-1}=4^3=64 \\\\ & a_5=4^{5-1}=4^4=256 \end{aligned}

This first term is 1. The next four terms of the sequence are 4, 16, 64, 256.

In order to find the explicit formula of an arithmetic sequence:

**Identify the first term.****Find the common difference for the sequence.****Write the explicit formula in the form \bf{\textbf{a}_\textbf{n}=\textbf{a}_1+\textbf{d}(\textbf{n}-1)} .**

Find the explicit formula for the sequence 5, 9, 13, 17, 21, ….

**Identify the first term.**

The first term is 5.

**Find the common difference for the sequence.**

The common difference is 4.

**Write the explicit formula in the form \bf{\textbf{a}_\textbf{n}=\textbf{a}_1+\textbf{d}(\textbf{n}-1)} .**

The explicit formula of this sequence is a_n=5+4(n-1).

Note: You can simplify this to the equation a_n=1+4 n.

Find the explicit formula for the sequence 3, 1, -1, -3, -5, ….

**Identify the first term.**

The first term is 3.

**Find the common difference for the sequence.**

The common difference is -2.

The explicit formula of this sequence is a_n=3+[-2(n-1)].

Note: You can simplify this to the equation a_n=5-2 n.

In order to find the explicit formula of a geometric sequence:

**Identify the first term.****Find the common ratio for the sequence.****Write the explicit formula in the form \bf{\textbf{a}_\textbf{n}=\textbf{a}_1(\textbf{r})^{\textbf{n}-1}} .**

Find the explicit formula for the sequence 1, 3, 9, 27, 81, ….

**Identify the first term.**

The first term is 1.

**Find the common ratio for the sequence.**

Each term is 3 times the one before it, so r=3.

**Write the explicit formula in the form \bf{\textbf{a}_\textbf{n}=\textbf{a}_1(\textbf{r})^{\textbf{n}-1}} .**

The explicit formula of this sequence a_n=1(3)^{n-1} \text { or } a_n=3^{n-1}.

Find the explicit formula for the sequence 4, 8, 16, 32, 64, ….

**Identify the first term.**

The first term is 4.

**Find the common ratio for the sequence.**

When the first term is not 1, you can solve for r for any given term.

For example, substitute values for the second term…

\begin{aligned}& a_n=a_1(r)^{n-1} \\\\ & 8=4(r)^{2-1} \\\\ & 8=4 r^1 \\\\ & 2=r\end{aligned}

The explicit formula of this sequence a_n=4(2)^{n-1}.

- Give students time to explore the graph of each type of sequence in explicit form. Specifically have them use different values for a_1 and d in arithmetic sequences, a_n=a_1+d(n-1), and different values for a_1 and r in geometric sequences, a_n=a_1(r)^{n-1}.

- Look for worksheets that include both arithmetic and geometric sequences. This requires students to practice identifying the type of sequence and remembering which general explicit formula to use.

**Confusing the recursive and explicit formulas for a sequence**

The recursive formula shows how to find the next term based on the previous term. The explicit formula shows how to find any term based on the relationships between the term number and the term itself. It is easy to confuse the two.

For example,

3, 6, 9, 12, 15, …

The recursive formula is a_{n+1}=a_n+3.

The explicit formula is a_n=3+3(n-1) or a_n=3 n.

**Thinking the difference between \bf{(\textbf{n}-1)} th and the nth term of the sequence is always positive**

The common difference is only positive if the sequence of numbers is increasing. In a decreasing sequence, the common difference is negative.

For example,

-2, \quad -4, \quad -6, \quad -8, \quad -10, …

Since it is decreasing by 2 each time, the explicit formula is -2-2(n-1) not -2+2(n-1) .

**Confusing exponents and subscript**

a_4 is not the same thing as a^4. Always pay attention to where the number is. A smaller number written above (superscript) is an exponent. In this case, a to the power of 4. A smaller number written below (subscript) is used to denote the number of the term number (or position of the term). In this case, the fourth term.

**Thinking sequences always start at \bf{\textbf{a}_1}**

While common, the first term of the sequence can be defined in other ways, such as a_0. Always pay close attention to how the specific terms of a are defined.

**Incorrectly simplifying geometric sequences**

In most cases, multiplying a_1 and r will change the outcomes of the sequence.

For example,

The explicit formula of this sequence a_n=4(2)^{n-1} is NOT the same as a_n=8^{n-1} .

- Number patterns
- Shape patterns
- Input/output tables
- Quadratic sequences
- Sequences

1. A sequence is defined by the explicit formula a_n=a_1+6(n-1) and a_1=-2.

Find the next four terms of the sequence.

4, 10, 16, 22

-8, -2, 4, 10

-12, -6, 0, 6

-12, -72, -432, -2,592

The formula is given in the question, a_n=a_1+6(n-1) and a_1=-2.

This creates the explicit formula a_n=-2+6(n-1).

Substitute the given value into the formula to calculate the new value.

\begin{aligned}& a_2=-2+6(2-1)=-2+6(1)=4 \\\\ & a_3=-2+6(3-1)=-2+6(2)=10 \\\\ & a_4=-2+6(4-1)=-2+6(3)=16 \\\\ & a_5=-2+6(5-1)=-2+6(4)=22 \end{aligned}

2. A sequence is defined by the explicit formula a_n=a_1(r)^{n-1} and a_1=3 and r=4.

Find the next four terms of the sequence.

12, 36, 108, 324

7, 11, 15, 19

7, 10, 13, 16

12, 48, 192, 768

The formula is given in the question, a_n=a_1(r)^{n-1} where a_1=3 and r=4.

This creates the explicit formula a_n=3(4)^{n-1}.

Substitute the given value into the formula to calculate the new value.

\begin{aligned}& a_2=3(4)^{2-1}=3 \cdot 4^1=12 \\\\ & a_3=3(4)^{3-1}=3 \cdot 4^2=48 \\\\ & a_4=3(4)^{4-1}=3 \cdot 4^3=192 \\\\ & a_5=3(4)^{5-1}=3 \cdot 4^4=768\end{aligned}

3. Find the explicit formula for the sequence 0.2, 0.5, 0.8, 1.1, 1.4, ….

a_n=0.3+0.2 n

a_n=0.2+0.3(n-1)

a_n=0.2+0.3 n

a_n=0.3+0.2(n-1)

The first term is 0.2, so a_1=0.2.

Find the common difference for the sequence.

The common difference is 0.3, so d=0.3.

Write the explicit formula in the form a_n=a_1+d(n-1).

The explicit formula of this sequence is a_n=0.2+0.3(n-1).

4. What is the explicit formula of the sequence?

-1, \quad -5, \quad -25, \quad -125, \quad -625, ….

a_n=5(-1)^{n-1}

a_n=(-5)^{n-1}

a_n=-1(5)^{n-1}

a_n=-5(1)^{n-1}

The first term is –1, so a_1=-1.

To find the common ratio for the sequence, use the general explicit formula to solve for r for a given term.

\begin{aligned} a_n&=a_1(r)^{n-1} \\\\ -5&=-1(r)^{2-1} \\\\ -5&=-1 \cdot r \\\\ 5&=r\end{aligned}

Write the explicit formula in the form a_n=a_1(r)^{n-1}.

The explicit formula of this sequence a_n=-1(5)^{n-1}.

5. Find the explicit formula for the following sequence:

\cfrac{1}{4}, \quad \cfrac{5}{8}, \quad 1, \quad 1 \cfrac{3}{8}, \quad 1 \cfrac{3}{4}

a_n=\cfrac{3}{8}+\cfrac{1}{4}(n-1)

a_n=\cfrac{3}{8}+\cfrac{1}{4} n

a_n=\frac{1}{4}+\frac{3}{8} n

a_n=\cfrac{1}{4}+\cfrac{3}{8}(n-1)

The first term is \cfrac{1}{4}, so a_1=\cfrac{1}{4}.

Find the common difference for the sequence.

The common difference is \cfrac{3}{8}, so d=\cfrac{3}{8}.

Write the explicit formula in the form a_n=a_1+d(n-1).

The explicit formula of this sequence is a_n=\cfrac{1}{4}+\cfrac{3}{8}(n-1).

6. What is the explicit formula for the given sequence?

\cfrac{1}{2}, \quad -1, \quad 2, \quad -4, \quad 8, ….

a_n=\cfrac{1}{2}(-2)^{n-1}

a_n=\cfrac{1}{2}(2)^{n-1}

a_n=(-1)^{n-1}

a_n=2\left(\cfrac{1}{2}\right)^{n-1}

The first term is \cfrac{1}{2}, so a_1=\cfrac{1}{2}.

To find the common ratio for the sequence, use the general explicit formula to solve for r for a given term.

\begin{aligned} a_n&=a_1(r)^{n-1} \\\\ -1&=\cfrac{1}{2}(r)^{2-1} \\\\ -1&=\cfrac{1}{2} \cdot r \\\\ -2&=r\end{aligned}

Look what happens if you solve for an odd numbered term, instead of the second term.

\begin{aligned} a_3&=\cfrac{1}{2}(r)^{3-1} \\\\ 2&=\cfrac{1}{2} \cdot r^2 \\\\ 4&=r^2\end{aligned}

Since the square root of 4 can be positive or negative, r could be positive or negative. To decide, look at the terms in the sequence. Since they alternate between positive and negative, r must be negative.

-4=r

Write the explicit formula in the form a_n=a_1(r)^{n-1}.

The explicit formula of this sequence is a_n=\cfrac{1}{2}(-2)^{n-1}.

No, the first term can be an integer, but it can also involve fractions or decimals or any real number.

A geometric sequence is a sequence of numbers that has a common ratio. The sum of the terms in a geometric sequence is a geometric series.

- Factoring
- Simultaneous equations
- Standard form

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