Question
Let $A_{1},{\text{ }A}_{2},{\text{ }A}_{3},\ldots\ldots..,A_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $A_{25},{\text{ }A}_{28},{\text{ }A}_{31}$ and $A_{36}$ is equal to:
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Solution
Sol. [Key Concept: Common difference $d = \frac{159 - 59}{40}$; then average $A_{25},A_{28},A_{31},A_{36}$.]
$${59{\text{ }A}_{1}{\text{ }A}_{2}{\text{ }A}_{3}\ldots{\text{ }A}_{39}159 }{d = \frac{159 - 59}{39 + 1} = \frac{5}{2} }$$Now, $A_{25} = 59 + 25\text{ }d = 121.5$
$$\begin{matrix} & A_{28} = 59 + 28d = 129 \\ & A_{31} = 59 + 31d = 136.5 \\ & A_{36} = 59 + 36d = 149 \end{matrix}$$
Mean of
$${A_{25},{\text{ }A}_{28},{\text{ }A}_{31},{\text{ }A}_{36} = \frac{121.5 + 129 + 136.5 + 149}{4} }{= \frac{536}{4} = 134 }$$
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