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If {A}_{1},{A}_{2},.....{A}_{31} are arithmetic means inserted

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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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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Text solutionVerified

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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Question Text	If $A_{1}, A_{2}, ..... A_{31}$ are arithmetic means inserted between the numbers $a$ and $b$ , then find the value of $(\frac{b + A _{51}}{b - A _{51}}) - (\frac{A _{1} + a}{A _{1} - a})$
Answer Type	Text solution:1
Upvotes	150