<s>[INST] Sequence proof if it exists For every integer r>=3 there exists a sequence $a_{1,\space }a_{2,}.....,a_r$ of nonzero integers with the property that
$a_1^2+a_2^2+....+a_{r-1}^2=a_r^2$
I tried to prove this with proof by induction (as to whether it was false or not) I'm not sure if that was the correct method. any help with solving this would be greatly appreciated
 [/INST] Let us start with the good old $3^2+4^2=5^2$, that is, $a_1=3$, $a_2=4$, and $a_3=5$, which gives a sequence of length $3$,  and produce a sequence of length $4$. 
The idea is to multiply the first two terms of our previous sequence  through by $3$, and add terms $4\cdot 5$ and $5\cdot 5$.  So for $r=4$ our sequence is $3\cdot 3$, $3\cdot 4$, $4\cdot 5$, and $5\cdot 5$. 
For a sequence of length $5$, multiply the first three terms of the previous sequence by $3$, and add $4\cdot 5\cdot 5$ and $5\cdot 5\cdot 5$. 
Continue.  
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