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Linear equations Substitution Arithmetic sequence Geometric sequence formulaHere you will learn about recursive formulas, including what they are and how to write them.
Students will first learn about recursive formulas as part of functions in high school.
The recursive formula is an equation that uses recursion to relate terms in a sequence. Recursion uses a rule over and over again. This relationship can be used to find the next term or the previous term.
For example,
Each number in a sequence is called a term, and each term is identified by its position within the sequence. For this sequence, the difference from term to term is common, making it an arithmetic sequence. You can write the first few terms as,
The first term, a_1=2
The second term, a_2=6
The third term, a_3=10
The fourth term, a_4=14
The fifth term, a_5=18
The n th term, a_n
The previous term, a_{n-1}
The next term, a_{n+1}
One way of generating this sequence would be to use a recursive formula, where each term is generated using the previous value.
When n=1, a_1=2.
When n=2, a_2=2+4=6.
When n=3, a_3=6+4=10, and so on.
So, the recursive formula is written as a_{n+1}=a_n+4.
The initial value, a_1, would need to be provided. The initial value could also be a_0.
Recursive formulas can be used to find particular solutions, but require starting at the beginning and repeatedly applying the rule to each term until the desired term is reached.
Another way of generating this sequence would be to use an explicit formula, which defines the relationship between the terms position and the value of the term itself.
To generate this sequence you can use the explicit formula for the sequence. For this sequence, the nth term would be a_n=2+4(n-1).
When n=1, a_1=2+4(0)=2.
When n=2, a_2=2+4(1)=6, and so on.
Step-by-step guide: Explicit formula
The following sequence of numbers is a special type of recursive sequence called the Fibonacci sequence.
1, 1, 2, 3, 5, 8, 13, 21, β¦The next number of the sequence can be found by adding the two previous terms together. Therefore, the Fibonacci numbers can be written using a recursive formula. You need to be given the first 2 numbers in the Fibonacci sequence.
F_{n+1}=F_{n-1}+F_n, \, F_1=1, \, F_2=1If you change the first 2 values, you can generate a different Fibonacci sequence.
For example, F_{n+1}=F_{n-1}+F_n, \, F_1=2, \, F_2=5
would give the sequence 2, 5, 7, 12, 19, 31, β¦
In general, a Fibonacci sequence can be written as, F_{n+1}=F_{n-1}+F_n, \, F_1=a, \, F_2=b
and would give the sequence a, \, b, \, a+b, \, a+2b, \, 2a+3b, \, 3a+5b, β¦
Did you know the Fibonnaci numbers occur often in nature? Flowers often have 3 or 5 or 8 petals. This is why a clover with 4 petals is difficult to find and is considered lucky.
How does this relate to high school math?
Use this quiz to check your grade 4 to 5 studentsβ 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 studentsβ 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 a recursive formula:
A sequence is defined by a recursive formula a_{n+1}=a_n-4 and has a_0=100.
Find the next four terms of the sequence.
The recursive formula is given in the question, a_{n+1}=a_n-4.
2Substitute the given initial value into the formula to calculate the new value, \bf{\textbf{a}_{\textbf{n}+1}}.
The initial value is given in the question, a_0=100.
Substituting this into a recursive formula gives a_1=a_0-4=100-4=96.
3Substitute the new value, \bf{\textbf{a}_{\textbf{n}+1}} , into the formula to calculate the next value.
a_2=a_1-4=96-4=924Repeat step \bf{3} until the desired number of terms has been generated.
\begin{aligned} & a_3=a_2-4=92-4=88 \\\\ & a_4=a_3-4=88-4=84 \end{aligned}The next four terms of the sequence are 96, 92, 88 and 84.
A sequence is defined by a recursive formula a_{n+1}=5 a_n and has a_0=1.
Find the next four terms of the sequence.
Find a recursive formula.
The recursive formula is given in the question, a_{n+1}=5 a_n
Substitute the given initial value into the formula to calculate the new value, \bf{\textbf{a}_{\textbf{n}+1}}.
The initial value is given in the question, a_0=1.
Substituting this into a recursive formula gives a_1=5 a_0=5 \times 1=5.
Substitute the new value, \bf{\textbf{a}_{\textbf{n}+1}} , into the formula to calculate the next value.
Repeat step \bf{3} until the desired number of terms has been generated.
The next four terms of the sequence are 5, 25, 125, 625.
In order to find a recursive formula of a sequence:
Find a recursive formula that satisfies the sequence 5, 8, 11, 14, 17, β¦
Find the arithmetic or geometric relationship linking the terms.
Each term in this sequence is 3 more than the one before it.
Write a recursive formula with correct notation.
The recursive formula is written as a_{n+1}=a_n+3.
Give one term of the sequence as well as a recursive formula.
The description needs to be more specific, so giving one term in the sequence, as well as a recursive formula is written as
a_{n+1}=a_n+3, a_1=5.
Write down a recursive formula which produces the sequence 3, 6, 12, 24, 48, β¦
Find the arithmetic or geometric relationship linking the terms.
Each term in this sequence is double the previous term.
Write a recursive formula with correct notation.
The recursive formula is written as a_{n+1}=2a_n.
Give one term of the sequence as well as a recursive formula.
The description needs to be more specific, so giving one term in the sequence, as well as a recursive formula is written as,
a_{n+1}=2 a_{n}, \, a_1=3.
Describe the sequence 32, 37, 42, 47, β¦ using a recursive formula.
Find the arithmetic or geometric relationship linking the terms.
Each term in this sequence equals the one before it, plus 5.
Write a recursive formula with correct notation.
The recursive formula is written as a_{n+1}=a_n+5.
Give one term of the sequence as well as a recursive formula.
The description needs to be more specific, so giving one term in the sequence, as well as a recursive formula is written as
a_{n+1}=a_n+5, \, a_1=32.
Write down a recursive formula which produces the sequence 20, β10, 5, β2.5, 1.25, β¦
Find the arithmetic or geometric relationship linking the terms.
Each term in this sequence is half the previous term. The terms also alternate between positive and negative.
Write a recursive formula with correct notation.
The recursive formula is written as a_{n+1}=-\cfrac{1}{2} \, a_n.
Give one term of the sequence as well as a recursive formula.
The description needs to be more specific, so giving one term in the sequence, as well as a recursive formula is written as
a_{n+1}=-\cfrac{1}{2} \, a_n, \, a_1=20.
1) A sequence is defined by the recursive formula a_{n+1}=a_n-4 and a_1=-2.
Find the next four terms of the sequence.
Substitute the given initial value into the formula to calculate the new value, Β a_{n+1}.
The initial value is given in the question, a_1=-2.
Substituting this into a recursive formula givesβ¦
\begin{aligned} & a_2=a_1-4=-2-4=-6 \\\\ & a_3=a_2-4=-6-4=-10 \\\\ & a_4=a_3-4=-10-4=-14 \\\\ & a_5=a_4-4=-14-4=-18 \end{aligned}
2) A sequence is defined by the recursive formula a_{n+1}=\cfrac{1}{4} \, a_n and a_o=1000.
Find the next four terms of the sequence.
The formula is given in the question, a_{n+1}=\cfrac{1}{4} \, a_n , where the initial value is a_o=1,000.
Substitute the given initial value into the formula to calculate the new value, Β a_{n+1}.
\begin{aligned} & a_1=\cfrac{1}{4} \, a_0=\frac{1}{4} \cdot 1000=250 \\\\ & a_2=\cfrac{1}{4} \, a_1=\frac{1}{4} \cdot 250=62.5 \\\\ & a_3=\cfrac{1}{4} \, a_2=\frac{1}{4} \cdot 62.5=15.625 \\\\ & a_4=\cfrac{1}{4} \, a_3=\frac{1}{4} \cdot 15.625=3.90625 \end{aligned}
3) Describe the sequence 1, 6, 11, 16, 21, … using a recursive formula.
Each term in this sequence equals the one before it, plus 5.
The recursive formula can be written as a_{n+1}=a_n+5.
One term in the sequence (typically the first), as well as a recursive formula is required for the description.
a_{n+1}=a_n+5, \, a_1=1.
4) Describe the sequence 34, 30, 26, 22, 18, … using a recursive formula.
Each term in this sequence equals the one before it, minus 4.
The recursive formula can be written as a_{n+1}=a_n-4.
One term in the sequence (typically the first), as well as a recursive formula is required for the description.
a_{n+1}=a_n-4, \, a_1=34
5) Describe the sequence 4, 8, 16, 32, 64, … using a recursive formula.
Each term in this sequence is double the previous term.
The recursive formula can be written as a_{n+1}=2 a_n.
One term in the sequence (typically the first), as well as a recursive formula is required for the description.
a_{n+1}=2 a_n, \, a_1=4
6) Describe the sequence -1, -1.5, -2.25, -3.375, -5.0625, … using a recursive formula.
Each term in this sequence is 1.5 times the previous term.
The recursive formula can be written as a_{n+1}=1.5 a_n.
One term in the sequence (typically the first), as well as a recursive formula is required for the description.
a_{n+1}=1.5 a_{n}, \, a_1=-1
No, the first term can be an integer and can also involve fractions or decimals or any real number.
This is another name for the recursive formula, which defines how the sequence changes from term to term.
A recurrence relation is an equation that identifies an initial condition (term before the other terms) and the rule for calculating the preceding terms. It is another name for the recursive formula of a recursive function.
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