I´m trying to learn something about matematical induction. This chapter starts with sums (Sigma)

There is this easy problem I do not understand.

It says calculate the Sigma: from k=1;to n. where $\displaystyle (1/k-1/(k+1))$

The textbook answer says: Answer: $\displaystyle 1-1/(n+1)$

But when I do it I get:

starts at k=1 so I get: $\displaystyle 1/1-1/(1+1) = 1/2$

Now I try to evaluate the last term:

$\displaystyle 1/n-1/(n+1)$ /multiply by n(n+1)

= $\displaystyle (n+1)/n(n+1) -n/(n(n+1)) $

that gives me $\displaystyle 1/(n^2+n)$

so I get $\displaystyle 1/2 +1/(n^2+n)$

Now to evaluate the sum I take the (first term +last term)multiply by n and devide by 2.

That gives me $\displaystyle (n(1/2+1(n^2+n))/2 $= $\displaystyle n/4+1/(2(n+1))$

clearly Iám doing someting very very stupid...anyone got an idéa whats wrong here?