Arithmetic and Geometric Sequences

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ARITHMETIC AND GEOMETRIC SEQUENCES

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Note: When you talk about an arithmetic sequence, the word arithmetic (in this context) is pronounced air-ith-ME-tic;
that is, the accent is on the third syllable.

DEFINITION (arithmetic sequence, common difference):
An arithmetic sequence is a sequence of the form un =un -1+d .
Here, d is called the common difference.

In an arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant.

EXAMPLE (arithmetic sequence):
The sequence 4,7,10 ,13,&ldots; is an arithmetic sequence. The common difference is 3 .
To go from term to term, you keep adding 3 .

CONNECTION BETWEEN ARITHMETIC SEQUENCES AND LINEAR FUNCTIONS:
Recall that linear functions graph as lines, and have a very special property:
equal changes in the input give rise to equal changes in the output.

Arithmetic sequences have this same special property:
equal changes in the input (e.g., moving from term to term)
give rise to equal changes in the output (determined by the common difference).

Thus, arithmetic sequences always graph as points along a line.
The graph of the sequence 4,7,10 ,13,&ldots; is shown below.

graph of an arithmetic sequence

EXAMPLE (arithmetic sequence):
The sequence 10,8,6 ,4,&ldots; is an arithmetic sequence. The common difference is -2 .
To go from term to term, you keep adding -2 (i.e., subtracting 2).

The graph of the sequence 10,8,6 ,4,&ldots; is shown below.

graph of an arithmetic sequence

DEFINITION (geometric sequence, common ratio):
A geometric sequence is a sequence of the form un =r⋅u n-1 .
Here, r is called the common ratio.

In a geometric sequence, each term is equal to the previous term, multiplied (or divided by) a constant.

EXAMPLE (geometric sequence):
The sequence 3,6,12 ,24,&ldots; is a geometric sequence. The common ratio is 2 .
To go from term to term, you keep multiplying by 2 .

CONNECTION BETWEEN GEOMETRIC SEQUENCES AND EXPONENTIAL FUNCTIONS:
There is a class of functions, called exponential functions, that have a very special property:
equal changes in the input cause the output to be successively multiplied by a constant.

Geometric sequences have this same special property:
equal changes in the input (e.g., moving from term to term)
cause the output to be successively multiplied by a constant (determined by the common ratio).

Thus, geometric sequences always graph as points along the graph of an exponential function.
The graph of the sequence 3,6,12 ,24,&ldots; is shown below.

graph of an arithmetic sequence

EXAMPLE (geometric sequence):
The sequence 100,50,25 ,12.5,&ldots; is a geometric sequence. The common ratio is 12 .
To go from term to term, you keep multiplying by 12 (i.e., dividing by 2).

Part of the graph of the sequence 100,50,25 ,12.5,&ldots; is shown below.

graph of an arithmetic sequence

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