Cfrgtkky

$$frac1{1times2} + frac1{2times3} + dots + frac1{n(n+1)} = frac1{n+1}$$

Prove for $n=1$: $$frac1{1times2}=frac1{1+1}=frac12$$

Hip: $$frac1{1times2} + frac1{2times3} + dots + frac1{n(n+1)} = frac1{n+1}$$

Demonstration: $$frac1{n+1} + frac1{(n+1)(n+2)}=dots=frac1{(n+1)+1}$$
My problem is that I can't find the correct algebra steps to get from the beginning of the demonstration to the end of the demonstration.

Actually, this is incorrect. The correct answer should be $1-frac1{n+1}$.

We can show that the above is true for $n=1$ easily. Now let us show $1-frac1{n+1}+frac1{(n+1)(n+2)}=1-frac1{n+2}$

We can prove that $frac1{n+1}-frac1{n+2}=frac {n+2}{(n+1)(n+2)}-frac {n+1}{(n+1)(n+2)}=frac{n+2-n-1}{(n+1)(n+2)}=frac1{(n+1)(n+2)}$ and vice versa. The $frac1{n+1}$ terms cancel out, giving us $1-frac1{n+2}$

$frac {1}{1times 2}=1-1/2$

$frac{1}{2times 3}=1/2 - 1/3$

........

$frac {1}{n(n+1)} =1/n- frac {1}{n+1}$

Add them up and cancel the middle terms to get $1-frac {1}{n+1}=frac {n}{n+1}$

You can see immediately that
what you have stated is false
because the left side increases
with increasing n
while the right side decreases.

So,
you have to find out
what you really want to prove.