4.2 Q-16

Exercise 4.2 — Question 16

Problem

Find the arithmetic mean between the following pairs of numbers:

(i) $\sqrt{2}$ and $3\sqrt{2}$

(ii) $\frac{1}{3}$ and $\frac{1}{5}$

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Key Concept: Arithmetic Mean

The arithmetic mean $A$ between two numbers $a$ and $b$ is the number inserted between them so that $a,A,b$ form an arithmetic sequence.

For an arithmetic sequence, the common difference must be constant: $A-a=b-A⟹2A=a+b⟹A=\frac{a+b}{2}$

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Solution

Part (i): 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 $=\sqrt{2}$ ✓

Part (ii): 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 $=\frac{4}{15}-\frac{1}{3}=\frac{4-5}{15}=-\frac{1}{15}$ ✓

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General Rule