Sequences of Real Numbers

Sequences of Real Numbers

Definition: A sequence of real numbers denoted $\{ a_n \}$ or $( a_n )_{n=1}^{\infty}$ is defined to be a function $f : \mathbb{N} \to \mathbb{R}$ such that $f(n) = a_n$. $a_n$ represents the $n^{\mathrm{th}}$ term of the sequence / the general term of the sequence.

For example, consider the sequence of natural numbers $(1, 2, 3, ... )$ whose function $f: \mathbb{N} \to \mathbb{N}$ is defined by the formula $f(n) = n$ for all $n \in \mathbb{N}$.

Another example of a sequence is $\left ( \frac{1}{n} \right )_{n=1}^{\infty} = \left ( 1, \frac{1}{2}, \frac{1}{3}, ... \right )$. which is also a function $f : \mathbb{N} \to \mathbb{R}$ such that $f(n) = \frac{1}{n}$ for all $n \in \mathbb{N}$.

In general a sequence can be described in three main ways:

  • A listing from an obvious pattern in the first few terms of the sequence, for example, the sequence of negative integers $(-1, -2, -3, ... )$.
  • By the general term of the sequence, for example, $\left ( n^2 \right)_{n=1}^{\infty} = ( 1, 4, 9, 16, 25, ... )$.
  • Recursively by defining each term in terms of earlier terms, for example, the Fibonacci sequence whose first and second terms $f_1 = f_2 = 1$, and for $n ≥ 3$, $f_n = f_{n-1} + f_{n-2}$, that is the sequence $(1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... )$.

Example 1

Determine the first ten terms of the sequence given that $a_1 = 1$, $a_2 = 2$, and for $n ≥ 3$, $a_n = a_{n-1}a_{n-2}$, and write a formula for the general term of this sequence.

We will place the terms in a table for organization purposes:

$a_1 = 1 = 2^0$
$a_2 = 2 = 2^1$
$a_3 = a_2a_1 = 2 = 2^1$
$a_4 = a_3a_2 = 4 = 2^2$
$a_5 = a_4a_3 = 8 = 2^3$
$a_6 = a_5a_4 = 32 = 2^5$
$a_7 = a_6a_5 = 256 = 2^8$
$a_8 = a_7a_6 = 8192 = 2^{13}$
$a_9 = a_8a_7 = 2097152 2^{21}$
$a_{10} = a_9a_8 = 17179869184 2^{34}$

Let $( f_n )$ denote the Fibonacci sequence and define $f_0 = 0$. Then $a_n = 2^{f_{n-1}}$ is a formula that represents the recursive sequence outlined in example 1. For example, $a_6 = 2^{f_5} = 2^5 = 32$.

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