j-james/math

Logo

All my math notes, now in Markdown.

View the Project on GitHub j-james/math

Sequences

Learning Targets

You should be able to

Concepts / Definitions

A sequence is an ordered set of numbers; it is a function with sequential natural numbers as the domain and the terms of the sequence as the range.

Sequence values (terms or output numbers) are written by using subscripts. The first term is $a_1$, the second term is $a_2$, and the $n$th term is $a_n$.

An explicit formula would have the form $a_n = \frac{3n}{n+1}$

Sequences can be finite or infinite.
Let $a_n$ be a sequence of real numbers, and let $L$ be a finite number. A sequence will converge if \(\lim_{x \to \infty} a_n = L\) If the sequence is infinite or nonexistant, the sequence diverges.

A recursive formula is a rule in which one or more previous terms are used to generate the next term.

An arithmetic sequence has an explicit rule or form $a_n = a_1 + (n-1)d$, where $d$ is the common difference between terms.
Recursive Rule: $a_n = a_{n-1} * r$ for $n \geq 2$

A geometric sequence has an explicit rule or form $a_n = a_1 r^{n-1}$, where $r$ is the common ratio between terms.
Recursive Rule: $a_n = a_{n-1} * r$ for $n \geq 2$

Fibonacci Sequence

\[a_1 = 1, \quad a_2 = 2, \quad a_n a_{n-2} + a_{n-1}\]

Fibonacci Sequence Golden Ratio

As the spiral continues, the ratios of the numbers approach a number called the Golden Ratio.

\[\frac{a+b}{a} = \frac{1+\sqrt{5}}{2} \approx 1.6180\]

Examples

Example 1

Write an explicit rule for $5, -7, 9, 11, 13, …$

Example 2

Determine if the following sequence converges or diverges. $2, \frac 32, \frac 43, \frac 54, …$

Example 3

Write an explicit rule for the following sequence. $3, 9, 17, 27, 39$

Example 4

Write an explicit rule for a geometric sequence given that $a_3 = 54$, and $a_6 = 2$

Exercises

Calculator okay, find with algebra.

  1. Determine if the following sequences converge or diverge.
    1. $\frac 12, \frac 14, \frac 18, \frac{1}{16}, …$
    2. $-1,1,-1,1,-1,…$
    3. $a_n = 1.3^{n-1}$
    4. $a_n = \frac{1-2n}{n+1}$
  2. Write an explicit and recursive rule for $-5, -2, 1, 4, …$
  3. Write an explicit rule for $\frac e2, \frac{e^2}{3}, \frac{e^3}{4}, …$
  4. Write an explicit rule for $\frac{-6}{5}, \frac{7}{15},\frac{-8}{45},\frac{1}{15},\frac{-2}{81}, …$
  5. Write an explicit rule for $2, 7, 16, 29, 46$
  6. Write an explicit rule $1\frac 12, 3\frac 14, 5\frac 16, 7\frac 18, …$
  7. Given the ninth term is -5 and the fifteenth term is 31, write an arithmetic explicit rule.
  8. Given $a_4 = 2$, and $a_7 = \frac{54}{125}$, write a geometric explicit rule.
  9. Find $x$ so that $x$, $x+2$, and $x+3$ are consecutive terms of a geometric sequence.
  10. Suppose that you have been hired at an annual salary of $30,000 and expect to receive annual increases of 3%. What will your salary be when you begin your fifth year?