examday

/TMUA

A sequence $(a_n)$ satisfies

for every positive integer $n$.

A student claims:

If $a_1$ is even, then every term of the sequence is even.

The student writes:

• 1. $a_1$ is even.
• 2. Assume $a_k$ is even.
• 3. Then $a_{k+1}=a_k+a_{k-1}$, which is the sum of two even numbers.
• 4. Therefore $a_{k+1}$ is even, completing the induction.

What is the main flaw?