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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
See also: Cubic graph
On this page, you will look specifically at finding the n th term for an arithmetic or geometric sequence.
Teaching explicit formulas? Use this quiz to check your grades 4 to 5 studentsβ understanding of number patterns. 10+ questions with answers covering a range of topics on number patterns to identify areas of strength and support!
DOWNLOAD FREETeaching explicit formulas? Use this quiz to check your grades 4 to 5 studentsβ understanding of number patterns. 10+ questions with answers covering a range of topics on number patterns to identify areas of strength and support!
DOWNLOAD FREEAn 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
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 = β4 (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
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, β¦ 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?
In order to generate a sequence using an explicit formula:
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.
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).
2Substitute the given value into the formula to calculate the new value.
a_2=3+2(2-1)=3+2(1)=53Repeat step \bf{2} until the desired number of terms has been generated.
\begin{aligned}& a_3=3+2(3-1)=3+2(2)=7 \\\\ & a_4=3+2(4-1)=3+2(3)=9 \\\\ & a_5=3+2(5-1)=3+2(4)=11\end{aligned}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.
Repeat step \bf{2} until the desired number of terms has been generated.
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:
Find the explicit formula for the sequence 5, 9, 13, 17, 21, β¦.
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, β¦.
Identify the first term.
The first term is 3.
Find the common difference for the sequence.
The common difference is -2.
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=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:
Find the explicit formula for the sequence 1, 3, 9, 27, 81, β¦.
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, β¦.
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 termβ¦
\begin{aligned}& a_n=a_1(r)^{n-1} \\\\ & 8=4(r)^{2-1} \\\\ & 8=4 r^1 \\\\ & 2=r\end{aligned}
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=4(2)^{n-1}.
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.
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.
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, β¦.
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, β¦.
The first term is β1, 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}
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, β¦.
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.
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