+118608 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
+118608 Excellent answer Heureka!
$$\frac{1*3*5*7......*99}{2*4*6*8*......*100}\\\\
\frac{1*3*5*7......*99}{1}\times \frac{1}{2*4*6*8*....*100}\\\\
\frac{100!}{2*4*6*...*100}\times \frac{1}{2*4*6*8*....*100}\\\\
\frac{100!}{2^{50}(1*2*3*4*....*50)}\times \frac{1}{2^{50}(1*2*3*4*....*50)}\\\\
\frac{100!}{2^{50}(50!)}\times \frac{1}{2^{50}(50!)}\\\\
\frac{100!}{2^{100}\times 50!\times 50!}\\\\$$
$${\frac{{\mathtt{100}}{!}}{\left({{\mathtt{2}}}^{\left({\mathtt{100}}\right)}{\mathtt{\,\times\,}}{\mathtt{50}}{!}{\mathtt{\,\times\,}}{\mathtt{50}}{!}\right)}} = {\mathtt{0.079\: \!589\: \!237\: \!600\: \!582\: \!7}}$$
$$0.08<\frac{1}{10}$$
+118608 Excellent answer Heureka!
$$\frac{1*3*5*7......*99}{2*4*6*8*......*100}\\\\
\frac{1*3*5*7......*99}{1}\times \frac{1}{2*4*6*8*....*100}\\\\
\frac{100!}{2*4*6*...*100}\times \frac{1}{2*4*6*8*....*100}\\\\
\frac{100!}{2^{50}(1*2*3*4*....*50)}\times \frac{1}{2^{50}(1*2*3*4*....*50)}\\\\
\frac{100!}{2^{50}(50!)}\times \frac{1}{2^{50}(50!)}\\\\
\frac{100!}{2^{100}\times 50!\times 50!}\\\\$$
$${\frac{{\mathtt{100}}{!}}{\left({{\mathtt{2}}}^{\left({\mathtt{100}}\right)}{\mathtt{\,\times\,}}{\mathtt{50}}{!}{\mathtt{\,\times\,}}{\mathtt{50}}{!}\right)}} = {\mathtt{0.079\: \!589\: \!237\: \!600\: \!582\: \!7}}$$
$$0.08<\frac{1}{10}$$
