Sequences and Factorials

Sequences Defined by a Recursive Formula

• Sequences appear in nautilus shells, pinecones, tree branches, and other natural structures.

• The Fibonacci sequence is a sequence where each term is the sum of the two preceding terms.

  • The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, …

  • Examples include the Calla Lily (1 petal), Black-Eyed Susan (13 petals), and daisies (21 or 34 petals).

• Each term depends on the terms that come before it.

• The Fibonacci sequence is best described using a recursive formula.

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

  • It consists of the value of an initial term (or terms) and an equation defining $a_n$ in terms of preceding terms.

Recursive Formula Example

• Given:

  • $a_1 = 3$

  • $an = 2a{n-1} - 1$, for $n ≥ 2$

• Finding subsequent terms:

  • $a_1 = 3$

Calculation of Terms

• $a2 = 2a1 - 1 = 2(3) - 1 = 5$

• $a3 = 2a2 - 1 = 2(5) - 1 = 9$

• $a4 = 2a3 - 1 = 2(9) - 1 = 17$

• The first four terms of the sequence are {3, 5, 9, 17}.

Fibonacci Sequence Recursive Formula

• The recursive formula states the first two terms and defines each successive term as the sum of the preceding two terms.

  • $a_1 = 1$

  • $a_2 = 1$

  • $a<em>n = a</em>{n-1} + a_{n-2}$, for $n ≥ 3$

• To find the tenth term, add the eighth and ninth terms:

  • $a<em>{10} = a</em>9 + a_8 = 34 + 21 = 55$

General Note: Recursive Formula

• A recursive formula defines each term of a sequence using preceding term(s).

• Recursive formulas must always state the initial term(s) of the sequence.

Q & A

• Must the first two terms always be given in a recursive formula?

  • No, many recursive formulas define each term using only one preceding term, requiring only the first term to be defined.

How To: Write the First n Terms of a Sequence Given a Recursive Formula

1. Identify the initial term, $a_1$, which is given as part of the formula.

2. To find the second term, $a<em>2$, substitute the initial term into the formula for $a</em>{n-1}$. Solve.

3. To find the third term, $a_3$, substitute the second term into the formula. Solve.

4. Repeat until you have solved for the $n^{th}$ term.

Example: Writing the Terms of a Sequence Defined by a Recursive Formula

• Write the first five terms of the sequence defined by the recursive formula:

  • $a_1 = 9$

  • $a<em>n = 3a</em>{n-1} - 20$, for $n ≥ 2$

Solution

• The first term is given.

• For each subsequent term, replace $a_{n-1}$ with the value of the preceding term.

  • $n = 1, a_1 = 9$

  • $n = 2, a<em>2 = 3a</em>1 - 20 = 3(9) - 20 = 27 - 20 = 7$

  • $n = 3, a<em>3 = 3a</em>2 - 20 = 3(7) - 20 = 21 - 20 = 1$

  • $n = 4, a<em>4 = 3a</em>3 - 20 = 3(1) - 20 = 3 - 20 = -17$

  • $n = 5, a<em>5 = 3a</em>4 - 20 = 3(-17) - 20 = -51 - 20 = -71$

• The first five terms are {9, 7, 1, -17, -71}.

Try It

• Write the first five terms of the sequence defined by the recursive formula:

  • $a_1 = 2$

  • $a<em>n = 2a</em>{n-1} + 1$, for $n ≥ 2$

• The first five terms are {2, 5, 11, 23, 47}.

Using Factorial Notation

• Factorial is a new operator denoted by "!".

• Factorial asks us to multiply together all the positive integers that come before the number in front of the symbol.

• Factorial is used heavily in Combinatorics, a branch of mathematics concerned with methods for counting sets.

• The formulas for some sequences include products of consecutive positive integers.

Definition of n Factorial

• $n$ factorial, written as $n!$, is the product of the positive integers from 1 to $n$.

  • For example:

    • $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$

    • $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$

• An example of a formula containing a factorial is $a_n = (n+1)!$.

  • The sixth term of the sequence can be found by substituting 6 for n:

    • $a_6 = (6+1)! = 7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5040$

• The factorial of any whole number $n$ is $n! = n(n-1)!$. Therefore, $5!$ can also be thought of as $5 \cdot 4!$.

A General Note: Factorial

• $n$ factorial is a mathematical operation that can be defined using a recursive formula.

• The factorial of $n$, denoted $n!$, is defined for a positive integer $n$ as:

  • $0! = 1$

  • $1! = 1$

  • $n! = n(n-1)(n-2)…(2)(1)$ , for $n ≥ 2$

• The special case $0!$ is defined as $0! = 1$.

Q & A

• Can factorials always be found using a calculator?

  • No, factorials get large very quickly—faster than even exponential functions! When the output gets too large for the calculator, it will not be able to calculate the factorial.

Example: Writing the Terms of a Sequence Using Factorials

• Write the first five terms of the sequence defined by the explicit formula $a_n = 5n(n+2)!$

Solution

• Substitute $n = 1, n = 2$, and so on in the formula.

  • $n = 1, a_1 = 5(1)(1+2)! = 5(3!) = 5 \cdot 3 \cdot 2 \cdot 1 = 30$

  • $n = 2, a_2 = 5(2)(2+2)! = 10(4!) = 10 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 240$

  • $n = 3, a_3 = 5(3)(3+2)! = 15(5!) = 15 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 1800$

  • $n = 4, a_4 = 5(4)(4+2)! = 20(6!) = 20 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 14400$

  • $n = 5, a_5 = 5(5)(5+2)! = 25(7!) = 25 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 126000$

• The first five terms are {30, 240, 1800, 14400, 126000}.

Try It

• Write the first five terms of the sequence defined by the explicit formula $a_n = \frac{n}{(n+1)!2^n}$

• The first five terms are {$\frac{1}{4}$, $\frac{1}{12}$, $\frac{1}{96}$,$\frac{1}{2880}$,$\frac{1}{8640}$}.