Let-m-and-n-be-two-positive-integers-satisfy-m-n-1-10-12-1-12-14-1-14-16-1-2012-2014-find-the-smallest-possible-value-of-m-n-

Question Number 96558 by bobhans last updated on 02/Jun/20

Let m and n be two positive integers

satisfy \frac{m}{n} = \frac{1}{10 \times 12} + \frac{1}{12 \times 14} + \frac{1}{14 \times 16} + \dots + \frac{1}{2012 \times 2014}

find the smallest possible value of

m + n

Answered by john santu last updated on 02/Jun/20

\Rightarrow \frac{m}{n} = \frac{1}{4} \underset{p = 5}{\sum^{1006}} \frac{1}{p (p + 1)}

= \frac{1}{4} \underset{p = 5}{\sum^{1006}} \frac{1}{p} - \frac{1}{p + 1} [telescopy]

= \frac{1}{4} (\frac{1}{5} - \frac{1}{1007}) = \frac{501}{10070}

since \gcd (501, 10070) = 1

we have m + n = 10571

Commented by selea last updated on 02/Jun/20

I think \frac{1}{4} has problem

Commented by john santu last updated on 02/Jun/20

why ?

Commented by bobhans last updated on 02/Jun/20

correct mr john . thanks

Commented by selea last updated on 02/Jun/20

\frac{1}{10 \times 12} + \frac{1}{12 \times 14} \dots + \frac{1}{2012 \times 2014} \neq \frac{1}{4} \underset{p = 5}{\sum^{1006}} (\frac{1}{p (p + 1)})

Commented by bobhans last updated on 02/Jun/20

\frac{1}{120} = \frac{1}{4 \times 5 \times 6} = \frac{1}{4} \times \frac{1}{5 \times 6}

\frac{1}{12 \times 14} = \frac{1}{4 \times 6 \times 7} = \frac{1}{4} \times \frac{1}{6 \times 7}

\frac{1}{14 \times 16} = \frac{1}{4 \times 7 \times 8} = \frac{1}{4} \times \frac{1}{7 \times 8}

Commented by bobhans last updated on 03/Jun/20

your wrong sir selea

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