4.2 Q-8

Exercise 4.2 — Question 8

Problem: Find the arithmetic mean between $\sqrt{2}$ and $3\sqrt{2}$.

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Key Concept: Arithmetic Mean

If $A$ is the arithmetic mean (A.M.) between two numbers $a$ and $b$, then $a$, $A$, $b$ form an arithmetic sequence. This requires:

$A-a=b-A$

Solving for $A$:

$2A=a+b⟹A=\frac{a+b}{2}$

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Solution

Here $a=\sqrt{2}$ and $b=3\sqrt{2}$.

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

Verification: The sequence $\sqrt{2},2\sqrt{2},3\sqrt{2}$ has a common difference of $\sqrt{2}$ ✓

$\boxed{A=2\sqrt{2}}$

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