Sequences & Series Question - None

Question

A sequence of real numbers $a_1, a_2, a_3, \ldots, a_{n+1}$ is such that $a_1 = 0$, $|a_2| = |a_1 + 1|$, $|a_3| = |a_2 + 1|, \ldots$

Then $a_{2020}$ cannot be equal to

Accepted Answer

Starting with $a_1 = 0$:

$|a_2| = |0+1| = 1$, so $a_2 = ±1$

Case 1: If $a_2 = 1$: $|a_3| = 2$, so $a_3 = ±2$

Case 2: If $a_2 = -1$: $|a_3| = 0$, so $a_3 = 0$

Pattern analysis: The sequence values oscillate between 0, 1, 2, –1, etc. The value –2 is never achieved.

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