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Paradoxica
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I will leave the base case for you to verify.
Inductive hypothesis:
Known information:
So we write out the definition of a_(n+1):
}{a_{n-1}}\text{ (Inductive hypothesis)}\\ &= \frac{1+3a_n a_{n-1}-a_n a_{n-2}}{a_{n-1}} \\ &= \frac{1+3a_n a_{n-1}- (1+a_{n-1}^2)}{a_{n-1}} \\ &= \frac{3a_n a_{n-1}-a_{n-1}^2}{a_{n-1}} \\ &= 3a_n-a_{n-1} \end{align*})
Hence, we have shown that if the statement is true for some integer n, it is also true for n+1.
By the principle of induction, it is true for all integers n≥2.
Inductive hypothesis:
Known information:
So we write out the definition of a_(n+1):
Hence, we have shown that if the statement is true for some integer n, it is also true for n+1.
By the principle of induction, it is true for all integers n≥2.