Question 17
An infinite sequence of positive numbers $$x_1,x_2,x_3,\ldots,x_n,x_{n+1},\ldots$$ satisfies $$x_n^2=(3n+7)+(n-3)x_{n+1}$$, where $$x_n$$ is the $$n^{\text{th}}$$ term of the sequence. Then the numerical value of $$x_1$$ is ______.
Correct Answer: 2
Solution
Put $$n=3$$ to get $$x_3^2=16$$, so positivity gives $$x_3=4$$. Next, putting $$n=2$$ gives $$x_2^2=13-x_3=9$$, so $$x_2=3$$. Finally, putting $$n=1$$ gives $$x_1^2=10-2x_2=4$$, hence $$x_1=2$$.
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