Exercise 4.2 — Question 5

Problem

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 $11$

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Key Concept: Arithmetic Mean

The arithmetic mean (A.M.) between two numbers $a$ and $b$ is:

$A.M.=\frac{a+b}{2}$

This is the single term that, when inserted between $a$ and $b$, forms an arithmetic sequence $a,A.M.,b$ with a constant common difference.

Why? In an A.P., the middle term equals the average of its neighbours: $A.M.-a=b-A.M.⟹2A.M.=a+b⟹A.M.=\frac{a+b}{2}$

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Solutions

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

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

Verification: The sequence $3,10,17$ has common difference $d=7$. ✓

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

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

Verification: The sequence $\frac{1}{2},1,\frac{3}{2}$ has common difference $d=\frac{1}{2}$. ✓

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

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

Verification: The sequence $-5,3,11$ has common difference $d=8$. ✓

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Summary