More interesting stuff

1. 1/6+1/12+1/20=?

=1/(2*3)+1/(3*4)+1/(4*5)
=1/2-1/3+1/3-1/4+1/4-1/5
=1/2-1/5
=3/10

You might ask: why 1/2*3=1/2-1/3?
let's solve 1/2-1/3 first
1/2-1/3=1/6  
you might notice that the solution is the difference of the two denominators over the product of the two denominators.
so let a=2 b=3
1/a-1/b=(b-a)/ab
a+1=b
so  [(a+1)-a]/a(a+1)
=1/a(a+1)=1/a-1/(a+1)
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2. 1/(3*5)+1/(5*7)+1/(7*9)=?

(reminder: 1/2(x)=x/2)

=1/2(1/3-1/5)+1/2(1/5-1/7)+1/2(1/7-1/9)
=1/2(1/3-1/5+1/5-1/7+1/7-1/9)
=1/2(1/3-1/9)
=1/2(2/9)
=1/9

Why 1/3*5=1/2*(1/3-1/5)?
Using the same method
1/2(1/3-1/5)
=1/2(2/15)
=1/15
let a=3 a+2=5
1/2[(a+2)-a/a(a+2)]
=1/2[2/a(a+2)]
let k=2
1/k[k/a(a+k)]

1/3-1/5=2/15
1/a-1/(a+2)
=(a+2)/a(a+2)-a/(a+2)
=2/a(a+2)
let k=2
k/a(a+k)=1/a-1/(a+k)
so 1/k[k/a(a+k)]
=1/k[1/a-1/(a+k)]=1/a(a+k)

test:
1/5*7=1/5(5+2)
a=5 k=2
=1/2(1/5-1/7)