Question
A sequence of real numbers $a_{1},a_{2},a_{3}\ldots..,a_{n + 1}$ is such that $a_{1} = 0,\left| a_{2} \right| = \left| a_{1} + 1 \right|,\left| a_{3} \right| = \left| a_{2} + 1 \right|,\ldots\ldots\ldots\left| a_{n + 1} \right| = \left| a_{n} + 1 \right|$. Then $\frac{a_{1} + a_{2} + \ldots\ldots + a_{n}}{n}$ can not be equal to
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Solution
Sol. [Key Concept: Iterate the recurrence to find the attainable values.]
$$a_{2} = a + 2a + 1$$
$$a_{3}^{2} = a_{2}^{2} + 2a_{2} + 1$$
$$a_{n + 1}^{2} = a_{n}^{2} + 2a_{n} + 1$$
Adding, $a_{n + 1}^{2} = a_{1}^{2} + 2\left( a_{1} + a_{2} + \ldots a_{n} \right) + n$
$${\Rightarrow \ a_{1} + a_{2} + \ldots\ldots + a_{n} = \frac{a_{n + 1}^{2} - n}{2} }{\Rightarrow \ \frac{\sum_{r = 1}^{n}\mspace{2mu} a_{r}}{n} = - \frac{1}{2} + \frac{a_{n + 1}^{2}}{2n} \geq - \frac{1}{2}}$$
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