<s>[INST] prove by induction: $3 + 5 + 7 + ... + (2n+1) = n(n+2)$ 
Use the principle of mathematical induction to prove that $$3 + 5 + 7 + ... + (2n+1) = n(n+2)$$ for all n in $\mathbb N$.

I have a problem with induction.  If anyone can give me a little insight it would be helpful.  
 [/INST] If you already know that
$$1+2+3+...+n=\frac{n(n+1)}2$$
we can try the following alternative approach:
$$3+5+7+\ldots+(2n+1)=$$
$$=1+2+3+4+5+\ldots+(2n+1)+(2n+2)-1-2(1+2+3+\ldots+ n+n+1)=$$
$$=\frac{(2n+2)(2n+3)}2-1-2\frac{(n+1)(n+2)}2=(n+1)(2n+3)-1-(n+1)(n+2)=$$
$$=(n+1)\left[2n+3-n-2\right]-1=(n+1)^2-1=n^2+2n=n(n+2)$$
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