<s>[INST] Guess the closed form on the following sequence? any help would be appreciated, have no idea where to start 
$u_1 = 2/3$ and $u_{k+1}$ such that: 
$$u_k + \frac{1}{(k+2)(k+3)}$$ for all, k are natural numbers 
guess a general formula (i.e the closed form) of the sequence
 [/INST] Hint: So $u_1=\frac{2}{3}$ and $u_2=\frac{2}{3}+\frac{1}{(3)(4)}$ and $u_3=\frac{2}{3}+\frac{1}{(3)(4)}+\frac{1}{(4)(5)}$ and so on.
Note that $\frac{1}{(3)(4)}=\frac{1}{3}-\frac{1}{4}$ and $\frac{1}{(4)(5)}=\frac{1}{4}-\frac{1}{5}$.
Do a couple more terms and notice the beautiful cancellations (telescoping). In general $\frac{1}{(k+2)(k+3)}=\frac{1}{k+2}-\frac{1}{k+3}$. 
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