Arithmetic Mean

If three numbers $a, A, b$ are in A.P. (in that order), then $A$ is called the arithmetic mean (A.M.) of $a$ and $b$. Being in A.P. means the common difference is the same on both sides: $A-a = b-A$. Solving for $A$:

So the arithmetic mean of two numbers is simply their ordinary average — the single number exactly midway between them on the number line.

Inserting several arithmetic means. More generally, $n$ numbers $A_1, A_2, \ldots, A_n$ are called $n$ arithmetic means between $a$ and $b$ if $a, A_1, A_2, \ldots, A_n, b$ together form an A.P. This combined list has $n+2$ terms in all, with $a$ as its first term and $b$ as its last (i.e. $(n+2)$th) term. Using the general-term formula of Section 2 with this A.P.'s own common difference $d$:

Once $d$ is known, each inserted mean is found by repeatedly adding $d$ to $a$:

Sum of the $n$ inserted means. The $n$ means $A_1,\ldots,A_n$ themselves form an A.P. (they are $n$ consecutive terms of the combined A.P.), so their sum can be found directly from the Section 2 sum formula — but there is a quicker route using symmetry. Pairing the $k$th mean from the start with the $k$th mean from the end:

$A_k + A_{n+1-k} = \big[a+kd\big] + \big[a+(n+1-k)d\big] = 2a+(n+1)d = a + \big[a+(n+1)d\big] = a+b,$ …