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Integration by Partial Fractions

If A1and A2 b...

Question

If $A_{1}$ and $A_{2}$ be the two A. M. s between two numbers $7$ and $1 / 7$ then $(2 A_{1} - A_{2}) (2 A_{2} - A_{1}) =$

$49$

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$48 / 7$

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$50 / 7$

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$1$

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Solution

The correct option is D
$1$

Find the value of $(2 A_{1} - A_{2}) (2 A_{2} - A_{1})$ :
Given $7, A_{1}, A_{2}, 1 / 7$ are in AP.
$\Rightarrow A_{1} - 7 = A_{2} - A_{1} \Rightarrow 2 A_{1} - A_{2} = 7 \dots . . . . . (i) a n d A_{2} - A_{1} = \frac{1}{7} - A_{2} \Rightarrow 2 A_{2} - A_{1} = 1 / 7 . . . \dots (i i)$
Multiply (i) and (ii). we get
$(2 A_{1} - A_{2}) (2 A_{2} - A_{1}) = 7 \times \frac{1}{7} = 1$
Hence, the correct option is (D).

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