Arithmetic Means

NOTE: The typical definition of the arithmetic mean of two numbers can be found in the following wiki: Arithmetic Mean. This entry is about an obscure usage of the term "arithmetic means."

Arithmetic means are the terms between two nonsuccessive terms of an arithmetic progression. The formula for the \(n^\text{th}\) term of an arithmetic progression can be used to find the arithmetic means between given terms of a progression.

In the arithmetic progression \(12,24,36,48,60,72\), what are the arithmetic means between \(24\) and \(72?\)

ANSWER

The arithmetic means between \(24\) and \(72\) are \(36,48,\) and \(60\). \(_\square\)

Insert five arithmetic means between \(13\) and \(-11\).

ANSWER

The arithmetic means will complete an arithmetic progression of \(7\) terms with \(a_1=13\) and \(a_7=-11\). First we shall find \(d\) from \(a_n=a_1+(n-1)d\) as follows:

\[\begin{align} a_7&=a_1+(n-1)d\\ -11&=13+(7-1)d\\ -11&=13+6d\\ 6d&=-24\\ d&=-4. \end{align}\]

Therefore, \(a_2=13-4=9,a_3=9-4=5,a_4=5-4=1,a_5=1-4=-3,\) and \(a_6=-3-4=-7\).

The desired arithmetic means are \(9,5,1,-3,\) and \(-7\). \(_\square\)