Arithmetic and Geometric Sequences, Given Values of Two Terms

A common problem found in IB mathematics textbooks (all levels), is to describe a sequence (either arithmetic or geometric) from two term values in the sequence. e.g. Determine the nth-term equation of a sequence given that the seventh term is -30 and the twelfth term is 20.  Basically describe the sequence:

20130720 -30to20

Put in another way:  How do we describe, generally, the arithmetic or geometric means (averages) between the given two values?

Usually, at the beginning of the chapter on sequences, the familiar nth-term equations for arithmetic and geometric versions are presented as:

20130720 EqArithmNth

for arithmetic sequences (d being the pattern, or “difference”) and

20130720 EqGeoNth

for geometric sequences (r being the pattern, or “ratio”)

Unfortunately, the sample question above doesn’t provide us with the first term, so we’ll have to do some algebra to derive some relationships.

For the arithmetic version of the problem:

20130720 EqArithmWorking1

so then filling in the gaps:

20130720 EqArithmWorking2

For the nth-term equation, we’ll need to solve for the first term:

20130720 EqArithmWorking3

For the geometric version of the problem:

20130720 EqGeoWorking1

so then filling in the gaps we get:

20130720 EqGeoWorking2

Observations:

  • Alternating signs (as expected when ratio is negative)
  • Values of powers can be considered an arithmetic sequence

For the nth-term equation, we’ll need to solve for the first term:

20130720 EqGeoWorking3

What happens if we were only provided two variables instead of two constants? How would derive the nth-term equation?   Well if this could accomplished, the equation could be quite useful if writing a computer algorithm.  Let’s say we were given a for the ath term and b for the bth term.  So,

20130720 EqAB

for arithmetic sequences:

20130720 EqABArtihmWorkinEDIT

(25 July 2013 Edit:  Removed original graphic due to arithmetic error)

for geometric sequences:

20130720 EqABGeoWorkinAren’t arithmetic and geometric means fun?

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This entry was posted in Mathematics, Sequences and Series and tagged , , . Bookmark the permalink.

One Response to Arithmetic and Geometric Sequences, Given Values of Two Terms

  1. navalator says:

    Fun to me is ice cream and sex.

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