4.2 Q-7

Exercise 4.2 — Question 7

Problem Statement

Find the arithmetic mean between the following pairs of numbers:

(i) $3$ and $17$ (ii) $\frac{1}{2}$ and $\frac{3}{2}$ (iii) $-5$ and $11$ (iv) $3+\sqrt{5}$ and $3-\sqrt{5}$

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Key Concept: Arithmetic Mean

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

Since the common difference must be constant: $A-a=b-A⟹2A=a+b⟹\boxed{A=\frac{a+b}{2}}$

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Solutions

Part (i): Between $3$ and $17$

$A=\frac{3+17}{2}=\frac{20}{2}=10$

Verification: $3,10,17$ — common difference $d=7$ ✓

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Part (ii): Between $\frac{1}{2}$ and $\frac{3}{2}$

$A=\frac{\frac{1}{2}+\frac{3}{2}}{2}=\frac{\frac{4}{2}}{2}=\frac{2}{2}=1$

Verification: $\frac{1}{2},1,\frac{3}{2}$ — common difference $d=\frac{1}{2}$ ✓

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Part (iii): Between $-5$ and $11$

$A=\frac{-5+11}{2}=\frac{6}{2}=3$

Verification: $-5,3,11$ — common difference $d=8$ ✓

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Part (iv): 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 $d=-\sqrt{5}$ ✓

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Summary Table