<s>[INST] Simplification of multiplying fractions Simplify $(1+\frac{1}{20\times22})(2+\frac{2}{21\times23})(2+\frac{2}{22\times24})(2+\frac{2}{23\times25})(13+\frac{13}{24\times26})$
I noticed the denominator of the five fractions are all in the form of:$$\frac{...}{(n-1)(n+1)}$$
Which can be simplified to:$$\frac{...}{n^2-1}$$
I futher attempted to move the integral part into the fraction, but it just got more complicated. How should I start from there?
 [/INST] $$m+\frac{m}{(n-1)(n+1)}=m\times(1+\frac{1}{(n-1)(n+1)})=m\times (\frac{n^2-1+1}{(n-1)(n+1)})=m\times\frac{n^2}{(n-1)(n+1)}$$
$$(1+\frac{1}{20\times22})(2+\frac{2}{21\times23})(2+\frac{2}{22\times24})(2+\frac{2}{23\times25})(13+\frac{13}{24\times26})=\frac{21^2}{20\times 22}\times 2\times \frac{22^2}{21\times 23}\times 2\times\frac{23^2}{22\times 24}\times 2\times \frac{24^2}{23\times 25}\times 13 \times \frac{25^2}{24\times 26}=105$$
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