Hi Chris I am just reposting it here where people will see it!!

I have referenced this puzzle address in the Sticky Topic "Puzzle" thread.

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Melody......Here's a problem that I ran into some time ago:

Show that

(1/2)*(3/4)*(5/6)*(7/8)*..............*(97/98) *(99/100) < (1/10)

It's "tricky" ......but simple.......all at the same time.

Maybe some of the members - or even, non-members - would like to try their hand at proving it !!

One word of warning.......your calculator won't do you any good on this one !!!

And ....as a "silly" problem...see if some of the users can figure THIS one out!!

Show that: sinx / n = 6

Melody
May 12, 2014

#2**+8 **

Excellent answer Heureka!

$$\frac{1*3*5*7......*99}{2*4*6*8*......*100}\\\\

\frac{1*3*5*7......*99}{1}\times \frac{1}{2*4*6*8*....*100}\\\\

\frac{100!}{2*4*6*...*100}\times \frac{1}{2*4*6*8*....*100}\\\\

\frac{100!}{2^{50}(1*2*3*4*....*50)}\times \frac{1}{2^{50}(1*2*3*4*....*50)}\\\\

\frac{100!}{2^{50}(50!)}\times \frac{1}{2^{50}(50!)}\\\\

\frac{100!}{2^{100}\times 50!\times 50!}\\\\$$

$${\frac{{\mathtt{100}}{!}}{\left({{\mathtt{2}}}^{\left({\mathtt{100}}\right)}{\mathtt{\,\times\,}}{\mathtt{50}}{!}{\mathtt{\,\times\,}}{\mathtt{50}}{!}\right)}} = {\mathtt{0.079\: \!589\: \!237\: \!600\: \!582\: \!7}}$$

$$0.08<\frac{1}{10}$$

Melody
May 12, 2014

#2**+8 **

Best Answer

Excellent answer Heureka!

$$\frac{1*3*5*7......*99}{2*4*6*8*......*100}\\\\

\frac{1*3*5*7......*99}{1}\times \frac{1}{2*4*6*8*....*100}\\\\

\frac{100!}{2*4*6*...*100}\times \frac{1}{2*4*6*8*....*100}\\\\

\frac{100!}{2^{50}(1*2*3*4*....*50)}\times \frac{1}{2^{50}(1*2*3*4*....*50)}\\\\

\frac{100!}{2^{50}(50!)}\times \frac{1}{2^{50}(50!)}\\\\

\frac{100!}{2^{100}\times 50!\times 50!}\\\\$$

$${\frac{{\mathtt{100}}{!}}{\left({{\mathtt{2}}}^{\left({\mathtt{100}}\right)}{\mathtt{\,\times\,}}{\mathtt{50}}{!}{\mathtt{\,\times\,}}{\mathtt{50}}{!}\right)}} = {\mathtt{0.079\: \!589\: \!237\: \!600\: \!582\: \!7}}$$

$$0.08<\frac{1}{10}$$

Melody
May 12, 2014