Chapter 11.CR, Problem 1CR

To determine

The difference between a sequence and series. Include in your discussion how to distinguish arithmetic, geometric, and Fibonacci-type sequence, and give the formulas for their general terms.

Explanation of Solution

The distinguishing a series from a sequence,

A sequences represent the list of numbers having a first term, a second term and so on and it is denoted as $s_n$. There are following example of sequence.

Arithmetic: A sequence with a common difference, $d$ and it is denoted as $a_n=a_1+\left(n-1\right)d$ Geometric: A sequence with a common ratio, $r$ and its denoted as $g_n=g_1{r}^{n-1}$.

Fibonacci: A sequence with first two term given, and subsequence terms the sum of the two previous terms. and its denoted as $s_n=s_{n-1}+s_{n-2}$, $n\ge 3$.

A series represent sum of term of a sequence $S_n$. There are following example of series.

Arithmetic series: Sum of the term of an arithmetic sequence:

$\begin{array}{rll}{A}_n& =& a_1+a_2+a_3+\cdots a_n\\ & =& n\left(\frac{a_1+a_n}{2}\right)\end{array}$

Geometric series: Sum of the term of an geometric sequence:

$\begin{array}{rll}{A}_n& =& g_1+g_2+g_3+\cdots g_n\\ & =& g_1\frac{\left(1-r^n\right)}{1-r},r\ne 1\end{array}$

Fibonacci series:

A sequence with first two terms given, and subsequence terms the sum of the two previous terms.

$s_n=s_{n-1}+s_{n-2},n\ge 3$