Arithmetic means

What you will be able to do

• find the arithmetic mean of two numbers, and test three numbers for an AP;
• insert any number of arithmetic means between two numbers;
• add up the inserted means without listing them;
• solve problems where the means meet a condition.

The mean of two numbers

The arithmetic mean (AM) of two numbers $a$ and $b$ is

It sits exactly half-way between them. So $a$, $A$, $b$ form an AP: $A - a = b - A$.

This gives a quick test. Three numbers $a$, $b$, $c$ are in an AP exactly when the middle one is the mean of the other two:

For example, $7$, $19$, $31$ are in an AP, since $2 \times 19 = 38 = 7 + 31$.

Inserting several means

Between two numbers $a$ and $b$ we can put $m$ numbers $A_1, A_2, \dots, A_m$ so that

is an AP. These numbers are the $m$ arithmetic means between $a$ and $b$.

The AP has $m + 2$ terms, with $a$ first and $b$ last. So $b = a + (m + 1)d$, and

For example, put $6$ means between $5$ and $33$. Then $d = \dfrac{28}{7} = 4$, and the means are $9, 13, 17, 21, 25, 29$.

The sum of the means

The means sit in an AP from $A_1$ to $A_m$, and $A_1 + A_m = a + b$, because they are the same distance from the ends. So

The $m$ means add up to $m$ times the single mean of $a$ and $b$. The $6$ means above add up to $6 \times 19 = 114$.

A worked example

Show that if $a$, $b$, $c$ are in an AP, then so are $b + c$, $c + a$ and $a + b$.

1. What we know. $2b = a + c$.
2. What we test. The middle number of the new list is $c + a$. Its double is $2c + 2a$.
3. The sum of the outer two. $(b + c) + (a + b) = 2b + a + c$.
4. Use $2b = a + c$. $2b + a + c = (a + c) + a + c = 2a + 2c$.

The two sides agree, so $b + c$, $c + a$, $a + b$ are in an AP. In fact, each is $a + b + c$ minus one of $a$, $c$, $b$, and subtracting from a constant keeps an AP.

Practise with new numbers

Try it in Python

The program inserts means between two numbers with exact fractions, prints the whole AP, and checks the sum of the means.

Common mistakes

• Dividing by $m$ instead of $m + 1$. The $m$ means make $m + 1$ steps from $a$ to $b$.
• Counting $a$ and $b$ as means. They are the ends. Only the numbers between them are the means.
• Testing an AP with $b^2 = ac$. That is the test for a geometric progression, in the next course. For an AP, use $2b = a + c$.
• Adding the means one by one. Use $m \times \dfrac{a + b}{2}$.

Remember these

• The AM of $a$ and $b$ is $\dfrac{a + b}{2}$.
• $a$, $b$, $c$ are in an AP exactly when $2b = a + c$.
• $m$ means between $a$ and $b$: $d = \dfrac{b - a}{m + 1}$, and $A_j = a + jd$.
• The $m$ means add up to $m \times \dfrac{a + b}{2}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the arithmetic mean of $-7$ and $23$.

2. Are $4$, $11$ and $19$ in an AP?

3. Insert $3$ arithmetic means between $10$ and $30$.

4. Twenty means are inserted between $1$ and $43$. What do they add up to?

5. If $x + 2$, $3x$ and $4x + 1$ are in an AP, find $x$.