Arithmetic means

The aver­age of two num­bers sits exactly halfway between them, so the three num­bers form an A.P. This aver­age is called the arith­metic mean. The same idea lets us fit any num­ber of evenly spaced val­ues between two given num­bers, which is called insert­ing arith­metic means. Both ideas are used con­stantly in later prob­lems.

Arith­metic mean of two num­bers#

If $a$, $A$, $b$ are in A.P., then $A$ is called the arith­metic mean (A.M.) of $a$ and $b$. By the prop­erty $2A = a + b$,

$$\displaystyle A = \frac{a + b}{2}$$

For exam­ple, the A.M. of 7 and 19 is $\displaystyle \frac{7 + 19}{2} = 13$, and indeed $7, 13, 19$ is an A.P. with com­mon dif­fer­ence 6.

Arith­metic mean of sev­eral num­bers#

More gen­er­ally, the arith­metic mean of $n$ num­bers $a_1, a_2, \ldots, a_n$ is their sum divided by $n$:

$$\displaystyle A = \frac{a_1 + a_2 + \cdots + a_n}{n}$$

This is the ordi­nary aver­age. For instance, the A.M. of 4, 9, 11 and 16 is $\displaystyle \frac{40}{4} = 10$.

Exam­ple 1#

The A.M. of two num­bers is 15 and one of them is 8. Find the other.

$\displaystyle \frac{8 + b}{2} = 15$ gives $8 + b = 30$, so $b = 22$.

Exam­ple 2#

Find the A.M. of $(x + y)^2$ and $(x - y)^2$.

$$\displaystyle \frac{(x + y)^2 + (x - y)^2}{2} = \frac{x^2 + 2xy + y^2 + x^2 - 2xy + y^2}{2} = \frac{2x^2 + 2y^2}{2} = x^2 + y^2$$

Insert­ing arith­metic means#

To insert $n$ arith­metic means between $a$ and $b$ means to find $n$ num­bers $A_1, A_2, \ldots, A_n$ such that

$$a, \ A_1, \ A_2, \ \ldots, \ A_n, \ b$$

is an A.P. This A.P. has $n + 2$ terms, with first term $a$ and last term $b$. So $b = a + (n + 1)d$, which gives the com­mon dif­fer­ence

$$\displaystyle d = \frac{b - a}{n + 1}$$

Then the means are $A_1 = a + d$, $A_2 = a + 2d$, and in gen­eral $A_k = a + kd$.

Exam­ple 3#

Insert 4 arith­metic means between 3 and 23.

Here $n = 4$, so $\displaystyle d = \frac{23 - 3}{4 + 1} = \frac{20}{5} = 4$. The means are

$$3 + 4 = 7, \quad 7 + 4 = 11, \quad 11 + 4 = 15, \quad 15 + 4 = 19$$

So the A.P. is $3, 7, 11, 15, 19, 23$.

Exam­ple 4#

Insert 5 arith­metic means between $-4$ and 14.

$\displaystyle d = \frac{14 - (-4)}{6} = \frac{18}{6} = 3$. The means are $-1, 2, 5, 8, 11$.

The sum of the inserted means#

Look back at Exam­ple 3. The four means add up to $7 + 11 + 15 + 19 = 52$, and $4 \times 13 = 52$, where 13 is the sin­gle A.M. of 3 and 23. This is no acci­dent. By Prop­erty 2 of an A.P., the means pair off from the two ends ($7 + 19$, $11 + 15$), and each pair adds up to $a + b$. So:

The sum of $n$ arith­metic means inserted between $a$ and $b$ is $n$ times their sin­gle A.M.:

$$\displaystyle A_1 + A_2 + \cdots + A_n = n \cdot \frac{a + b}{2}$$

Exam­ple 5#

Some arith­metic means are inserted between 2 and 38, and their sum is 200. How many means are there?

The sum of $n$ means is $\displaystyle n \cdot \frac{2 + 38}{2} = 20n$. So $20n = 200$ and $n = 10$.

Exam­ple 6#

$n$ arith­metic means are inserted between 7 and 71, and the 5th mean is 27. Find $n$.

The com­mon dif­fer­ence is $\displaystyle d = \frac{71 - 7}{n + 1} = \frac{64}{n + 1}$. The 5th mean is $A_5 = 7 + 5d = 27$, so $5d = 20$ and $d = 4$. Then $\displaystyle \frac{64}{n + 1} = 4$ gives $n + 1 = 16$, so $n = 15$.

Exam­ple 7#

Between 2 and 41, $m$ arith­metic means are inserted so that the ratio of the 4th mean to the $(m - 2)$th mean is $7 : 16$. Find $m$.

Here $\displaystyle d = \frac{39}{m + 1}$. The 4th mean is $2 + 4d$ and the $(m - 2)$th mean is $2 + (m - 2)d$. So

$$\displaystyle \frac{2 + 4d}{2 + (m - 2)d} = \frac{7}{16}$$

Sub­sti­tute $\displaystyle d = \frac{39}{m + 1}$ and mul­ti­ply the top and bot­tom by $m + 1$:

$$\displaystyle \frac{2(m + 1) + 156}{2(m + 1) + 39(m - 2)} = \frac{7}{16} \implies \frac{2m + 158}{41m - 76} = \frac{7}{16}$$

Cross-mul­ti­ply­ing, $32m + 2528 = 287m - 532$, so $255m = 3060$ and $m = 12$. Check: $d = 3$, the 4th mean is 14 and the 10th mean is 32, and $14 : 32 = 7 : 16$.

Prac­tice#

1. Find the A.M. of 12 and 30.
2. Find the A.M. of 6, 10, 15, 21 and 28.
3. The A.M. of two num­bers is 20 and one of them is 27. Find the other.
4. Find the A.M. of $a + 2b$ and $a - 2b$.
5. Insert 3 arith­metic means between 5 and 21.
6. Insert 4 arith­metic means between 10 and $-10$.
7. Insert 6 arith­metic means between 1 and 22.
8. Find the sum of 8 arith­metic means inserted between 4 and 30.
9. Some arith­metic means are inserted between 3 and 45, and their sum is 264. How many are there?
10. $n$ arith­metic means are inserted between 2 and 50, and the 3rd mean is 11. Find $n$.

Answers#