How do we simplify this? Confused. $$\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)\cdots\left(1-\frac{1}{n}\right)$$
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Why didn't you even try for small numbers? If you had, you could have canceled out all the terms. ;)
The expression turns out to be
$\dfrac{2}{3}.\dfrac{3}{4}.\dfrac{4}{5} \dots\dfrac{n-2}{n-1}.\dfrac{n-1}{n}$
Cancel out the terms and you're left with just $\frac{2}{n}$.
John Gowers
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Inceptio
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Wait, small numbers? – Mar 18 '13 at 08:04
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Yeah! Till $5$ or $6$ maybe?. You get $2/5$ for $n=5$ and $2/6$ for $n=6$. ;) – Inceptio Mar 18 '13 at 08:10
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you know $\frac{2}{3}<\frac{3}{4}<\cdots$ right? – Mar 18 '13 at 08:12