Equation: 1/(1*2) + 1/(2*3) + 1/(2*4) + ... + 1/(n*(n+1)) = n/(n+1)
I need help with solving this equation. I cannot seem to be able to do anything with (n+1)/(n+2) that I get on the right side after induction step.
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Hint $1$
$$\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}$$
write every term like above and sum.
Hint $2$
If you really need use induction then use the standard approach for $n+1$:
$$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{n(n+1)}+\frac{1}{(n+1)\cdot (n+2)}=\frac{n}{n+1}+\frac{1}{(n+1)\cdot (n+2)}$$
Finish the calculation and get
$$\frac{n+1}{n+2}$$
Arnaldo
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I really need to use induction and I got (n+1)/(n+2), but does that by itself prove anything or is there more to be done? It doesn't seem finished. – Pawel Jezierski Dec 19 '16 at 19:22
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That's the last and most important step for finite induction. In order to finish you have to prove that the statement is true for $n=1$ and assume that is true for $n$. – Arnaldo Dec 19 '16 at 19:24