Exercise 4.2 — Question 2

Arithmetic Mean Between Two Numbers

The arithmetic mean (AM) between two numbers $a$ and $b$ is the value $A$ such that $a,A,b$ form an arithmetic sequence.

Since consecutive terms of an AP have a constant common difference: $A-a=b-A⟹2A=a+b⟹A=\frac{a+b}{2}$

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Worked Examples

Part (i): Find the arithmetic mean between $\sqrt{2}$ and $3\sqrt{2}$

$A=\frac{\sqrt{2}+3\sqrt{2}}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$

Verification: $\sqrt{2},2\sqrt{2},3\sqrt{2}$ — common difference $=\sqrt{2}$ ✓

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Part (ii): Find the arithmetic mean between $\frac{1}{3}$ and $\frac{1}{5}$

$A=\frac{\frac{1}{3}+\frac{1}{5}}{2}=\frac{\frac{5+3}{15}}{2}=\frac{\frac{8}{15}}{2}=\frac{8}{30}=\frac{4}{15}$

Verification: $\frac{1}{3},\frac{4}{15},\frac{1}{5}$ — common difference $=\frac{4}{15}-\frac{5}{15}=-\frac{1}{15}$ ✓

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Part (iii): Find the arithmetic mean between $3+\sqrt{5}$ and $3-\sqrt{5}$

$A=\frac{(3+\sqrt{5})+(3-\sqrt{5})}{2}=\frac{6}{2}=3$

Verification: $3+\sqrt{5},3,3-\sqrt{5}$ — common difference $=-\sqrt{5}$ ✓

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Part (iv): Find the arithmetic mean between $a+b$ and $a-b$

$A=\frac{(a+b)+(a-b)}{2}=\frac{2a}{2}=a$

Verification: $a+b,a,a-b$ — common difference $=-b$ ✓

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Key Formula

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

where $A$ is the single arithmetic mean inserted between $a$ and $b$.