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:

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:

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

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:

so then filling in the gaps:

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

For the geometric version of the problem:

so then filling in the gaps we get:

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:

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,

for arithmetic sequences:

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

for geometric sequences:

Aren’t arithmetic and geometric means fun?