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Saturday, July 16, 2022

2022/052) Simplify expression (\frac{1}{2^3-1} + \frac{1}{2}) (\frac{1}{3^3-1} + \frac{1}{2}) \cdots (\frac{1}{100^3-1} + \frac{1}{2})

We have let t_n =   \frac{1}{t^3-1} + \frac{1}{2}

= \frac{1}{2}\frac{t^3+1}{t^3-1}

=\frac{1}{2}\frac{(t+1)(t^2-t+1)}{(t-1)(t^2+t+1)}

Now given value = \prod_{n=2}^{100} \frac{1}{2}\frac{(t+1)(t^2-t+1)}{(t-1)(t^2+t+1)}

= \frac{1}{2^{99}} \prod_{n=2}^{100} \frac{(t+1)}{(t-1)}\frac{(t^2-t+1)}{(t^2+t+1)}

now let us compute the 2 products 

\prod_{n=2}^{100} \frac{(t+1)}{(t-1)}= \frac{3}{1}.\frac{4}{2}.\frac{5}{3}\cdots \frac{98}{96}. \frac{99}{97}. \frac{100}{98}. \frac{101}{99}

In the above product the numerator of n^{th} term is same as denoniator of n+2^{nd} term giving

\prod_{n=2}^{100} \frac{(t+1)}{(t-1)} = \frac{100 * 101}{1 * 2}\cdots(1)

let us compute 2nd product

  \prod_{n=2}^{100} \frac{(t^2-t+1)}{(t^2+t+1)}= \frac{3}{7} ( \frac{ 7}{13} \cdots \frac{9901}{10101}

In the above product the numerator of n^{th} term is same as denoniator of n-1^{st} term giving

  \prod_{n=2}^{100} \frac{(t^2-t+1)}{(t^2+t+1)}= \frac{3}{10101}\cdots(2)

using (1) and (2) we get the required result

= \frac{1}{2^{99}}\frac{3}{10101} * 5050= \frac{2525}{2^{98} * 3367}

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