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Number Sequences

Investigating Sequences

A number sequence is a set of numbers defined by a rule that is valid for all positive integers.

For each number sequence, find the next three terms (separated by commas) as well as the

$100^{th}$ term.
For example:

$1,2,3,4,5,\fbox{6,7,8}\ldots\fbox{100}$

$2,4,6,8,10,$

$\ldots$

$-4,-1,2,5,8,$

$\ldots$

$100,89,78,67,56,$

$\ldots$

$1,4,16,64,256,$

$\ldots$ *Write the 100th term in the form

$a^b$

$1,-3,9,-27,81,$

$\ldots$ *Write the 100th term in the form

$\left(-a\right)^b$

$9,109,209,309,409,$

$\ldots$

$2,6,12,20,30,$

$\ldots$

$3,8,15,24,35,$

$\ldots$

$1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},$

$\ldots$ *Write the 100th term in the form

$\frac{a}{b^c}$

$4,16,36,64,100,$

$\ldots$

$4,20,64,176,448,$

$\ldots$ *You don't need to find the 100th term

$\frac{1}{2},\frac{1}{2},\frac{3}{8},\frac{1}{4},\frac{5}{32},$

$\ldots$ *You don't need to find the 100th term

$1,1,2,3,5,8,13,21,$

$\ldots$ *You don't need to find the 100th term

[challenge]

$2,6,7,5,0,-8,-19,$

$\ldots$ *You don't need to find the 100th term

[challenge]

$2,3,3,5,10,13,39,43,172,177,$

$\ldots$ *You don't need to find the 100th term

[challenge]

$1,11,21,1211,111221,312211,13112221,$

$\ldots$ *You don't need to find the 100th term

Types of Sequences

In an arithmetic sequence, each term is found by adding or subtracting the same number to the previous term.
In a geometric sequence, each term is found by multiplying or dividing the same number to the previous term.
For each of the following sequences, identify if it is an “arithmetic” or “geometric” sequence or “neither”.

$4,11,18,25,32,...$

$5,10,20,40,80,...$

$2,3,5,6,8,9,...$

$32,16,8,4,2,1,...$

$1,4,9,16,25,36,...$

$-1,1,-1,1,-1,1,...$

$2,1\frac{3}{7},\frac{6}{7},\frac{2}{7},-\frac{2}{7},...$

$0.1,0.01,0.001,0.0001,0.00001,...$