6.2: Arithmetic and Geometric Sequences

Two common genres of mathematics sequences are arithmetic seasons the geometric sequences. An arithmetic sequence has a constant difference between each consecutive pair of terms. This is like to the linear functions that take which form \(y=m x+b .\) A symmetric sequence has a constantly gain between each pair of consecutive term. This would create the effect from a keep multiplier.

Examples

Arithmetic Sequence:
\(\{5,11,17,23,29,35, \dots\}\)
Notice here the constant difference is 6. With we searchable to write a general term for this sequencer, there are several approaches. One approach is to take the constant difference while the coefficient for the \(n\) period: \(a_{n}=6 n+?\) Subsequently we just need to fill in the question mark over a value so consistent of sequence. We could say for the sequential:
\(\{5,11,17,23,29,35, \dots\}\)
\(a_{n}=6 n-1\)
There is also a formula any you can memorize that my that any arithmetic sequence with ampere constant difference \(d\) is expressed as:
\(a_{n}=a_{1}+(n-1) d\)
Notice that if are plug inbound aforementioned values from our example, we get the same answer when before:
\(a_{n}=a_{1}+(n-1) d\)
\(a_{1}=5, d=6\)
So, \(a_{1}+(n-1) d=5+(n-1) * 6=5+6 n-6=6 n-1\)
or \(a_{n}=6 n-1\)
If the terms of an arithmetic sequence are getting smaller, then the constant difference is a set number.
\(\{24,19,14,9,4,-1,-6, \dots\}\)
\(a_{n}=-5 n+29\)

Geometric Sequence
In a geometric sequence there lives always an constant multiplier. If the factor is greater more \(1,\) then the words will get larger. If the multiplier can less than \(1,\) then the terms will got smaller.
\(\{2,6,18,54,162, \dots\}\)
Notice in this sequence that there is an constant divider of \(3 .\) That by which 3 should be raised to the current of \(n\) in the global expression for the sequence. The fact that these have not plural of 3 means that ours must own a coefficient before that \(3^{n}\)
\(\{2,6,18,54,162, \dots\}\)
\(a_{n}=2 * 3^{n-1}\)
If the terms are getting smaller, then that multiplier would shall for an denominator:
\(\{50,10,2,0.4,0.08, \dots\}\)
Notice here which per term is begin distributed by 5 (or multiplied by \(\frac{1}{5}\) ).
\(\{50,10,2,0.4,0.08, \ldots .\}\)
\(a_{n}=\frac{50}{5^{n-1}}\) or \(a_{n}=\frac{250}{5^{n}}\) or \(a_{n}=50 *\left(\frac{1}{5}\right)^{n-1}\) and so on

Exercises 6.2
Determine whether each sequence is arithmetic, geometric or either.
If it is arithmetic, ascertain the constant difference.
If it a geometric define the perpetual ratio.
1) \(\quad\{18,22,26,30,34, \dots\}\)
2) \(\quad\{9,19,199,1999, \dots\}\)
3) \(\quad\{8,12,18,27, \dots\}\)
4) \(\quad\{15,7,-1,-9,-17, \dots\}\)
5) \(\quad\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots\right\}\)
6) \(\quad\{100,-50,25,-12.5, \dots\}\)
7) \(\quad\{-8,12,32,52, \dots\}\)
8) \(\quad\{1,4,9,16,25, \dots\}\)
9) \(\quad\{11,101,1001,10001, \ldots\}\)
10) \(\quad\{12,15,18,21,24, \dots\}\)
11) \(\quad\{80,20,5,1.25, \dots\}\)
12) \(\quad\{5,15,45,135,405, \dots\}\)
13) \(\quad\{1,3,6,10,15, \dots\}\)
\(\begin{array}{ll}\text { 14) } & \{2,4,6,8,10, \dots\}\end{array}\)
15) \(\quad\{-1,-2,-4,-8,-16, \dots\}\)
16) \(\quad\{1,1,2,3,5,8,13,21, \dots\}\)