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Arithmetic Mean

If A1,A2,.....

Question

If $A_{1}, A_{2}, . . . . . A_{31}$ are arithmetic means inserted between the numbers $a$ and $b$ , then find the value of $(\frac{b + A_{51}}{b - A_{51}}) - (\frac{A_{1} + a}{A_{1} - a})$

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Solution

We know that

$\Rightarrow t_{n} = a + (n - 1) d$

$\Rightarrow b = a + (n - 1) d$

$\Rightarrow d = \frac{b - a}{n + 1}$

We have

$\Rightarrow A_{1} = a + d = a + (\frac{b - a}{n + 1})$

$\Rightarrow A_{2} = a + 2 d = a + 2 (\frac{b - a}{n + 1})$

$\Rightarrow A_{n} = a + n d = a + n (\frac{b - a}{n + 1})$

$\Rightarrow A_{1} = a + d = a + (\frac{b - a}{51 + 1}) = a + (\frac{b - a}{52}) = \frac{b + 51 a}{52}$

$\Rightarrow A_{51} = a + d = a + 51 (\frac{b - a}{51 + 1}) = a + 51 (\frac{b - a}{52}) = \frac{a + 51 b}{52}$

Now, $(\frac{b + A_{51}}{b - A_{51}}) - (\frac{A_{1} + a}{A_{1} - a})$

$\Rightarrow (\frac{b + \frac{a + 51 b}{52}}{b - \frac{a + 51 b}{52}}) - (\frac{\frac{b + 51 a}{52} + a}{\frac{b + 51 a}{52} - a})$
On simplifying we get

$\Rightarrow \frac{103 b + a}{b - a} - \frac{b + 103 a}{b - a}$

$\Rightarrow \frac{103 b + a - b - 103 a}{b - a}$

$\Rightarrow \frac{102 b - 102 a}{b - a} = 102$

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