The number in the middle
If a, b, c are in AP, then b = (a + c)/2.
Halfway
The numbers 4, b, 16 are in AP. Where on the number line does b stand?
In an AP the same step is taken from 4 to b as from b to 16.
Where is b?
What this lesson covers
The idea
If a, b, c are in AP, then b = (a + c)/2, and b is called the arithmetic mean of a and c.
Halfway
The numbers 4, b, 16 are in AP. Where on the number line does b stand?
In an AP the same step is taken from 4 to b as from b to 16.
Where is b?
- Exactly halfway between 4 and 16
- Closer to 4
- Closer to 16
Put b in the middle
a and c stay still. Move b until the two gaps, b − a and c − b, are equal. Do it for all three pairs.
Arithmetic mean
If a, b, c are in AP, then b − a = c − b, so b = (a + c)/2. The number b is called the arithmetic mean of a and c.
b is exactly halfway between a and c: add the two numbers and halve. It is the ordinary average of a and c.
Notes
If a, b, c are in AP, then b = (a + c)/2, and b is the arithmetic mean of a and c.
Check yourself
Find the arithmetic mean of 10 and 24.
Answer: 17
(10 + 24)/2 = 34/2 = 17.
The numbers −5, b, 9 are in AP. Find b.
Answer: 2
b = (−5 + 9)/2 = 4/2 = 2.
The numbers x, 12, 20 are in AP. Find x.
Answer: 4
12 = (x + 20)/2, so x + 20 = 24 and x = 4. Check: 4, 12, 20 has d = 8.
Which three numbers are in AP?
- 2, 6, 10 — correct. Yes! The middle number 6 is (2 + 10)/2.
- 2, 6, 12. The middle of 2 and 12 would be 7, not 6.
- 2, 7, 10. The middle of 2 and 10 would be 6, not 7.