14 sticks

I was cleaning up my attic recently and found a set of at least 14 sticks which a curious Italian sold me some years ago. Trying hard to figure out why I bought it from him, I realized that the set has the incredible property that there are no three sticks that can form a triangle. If the set has two sticks of length 1, which are the smallest, what is the least possible length of the 14th stick?

Correct answer:

x₁₄ =  377

Step-by-step explanation:

$$\begin{gathered} x_1=1 \\ {x}_2=1 \\ {x}_3=x_1+x_2=1+1=2 \\ {x}_4=x_3+x_2=2+1=3 \\ {x}_5=x_4+x_3=3+2=5 \\ {x}_6=x_5+x_4=5+3=8 \\ {x}_7=x_6+x_5=8+5=13 \\ {x}_8=x_7+x_6=13+8=21 \\ {x}_9=x_8+x_7=21+13=34 \\ {x}_{10}=x_9+x_8=34+21=55 \\ {x}_{11}=x_{10}+x_9=55+34=89 \\ {x}_{12}=x_{11}+x_{10}=89+55=144 \\ {x}_{13}=x_{12}+x_{11}=144+89=233 \\ {x}_{14}=x_{13}+x_{12}=233+144=377 \end{gathered}$$

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