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Completing Geometric Sequences

Date: 02/16/98 at 17:55:27
From: James
Subject: Geometric Sequence

How do you find missing terms in a series of geometric sequences?

For example: what are the second and third terms of a geometric 
sequence that has a first term of -3 and a fourth term of -8/9?

Date: 02/16/98 at 20:16:36
From: Doctor Sam
Subject: Re: Geometric Sequence

Hi James.

A geometric sequence is formed by multiplying each term by the same 
value. For example,

    1, 5, 25, 125, ...
    8, 4, 2, 1, ...   

In the first case you multiply by 5 to get the next term, and in the 
second example you multiply by 1/2.

You ask how to insert terms into a geometric sequence.  For example, 
insert three numbers between 2 and 12 to make a geometric sequence.

Any geometric sequence can be represented algebraically by 

    a, ar,  ar^2, ar^3, ar^4, ...

where a is the first term and r is the number that multiplies each 
term to get the next.

In my example, a = 2, and three terms are missing before we have 12, 
so the sequence is:

    2,  2r,   2r^2,   2r^3,   12, ...

You don't really need to write out all the missing terms, but it may 
help you to think about the question.  In this case 12 = 2r^4. This is 
an equation that you can use to solve for r and, since you already 
know the first term, a, you can write down all the terms of the 

In general, you may be asked to insert 20 or 100 or n terms in between 
two given numbers. Just think through the pattern above and you will 
see that the last number is always ar^k, where 

    a = first number, 
    r = unknown factor, and 
    k = one more than the number of terms you want to insert.

Does that help?

-Doctor Sam,  The Math Forum
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Associated Topics:
High School Sequences, Series

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