Question

Sol. [Key Concept: Integer-sequence recurrence ⇒ closed form and term values.]

$$\begin{matrix} a_{n} + a_{n - 1} & \ = 2n \\ a_{n - 1} + a_{n - 2} & \ = 2(n - 1) \\ \Rightarrow \ a_{n} - a_{n - 2} & \ = 2 \\ a_{1},a_{2},a_{3},\ldots\ldots & \ = 100, - 96,102, - 94,104, - 92,\ldots \\ a_{n} & \ = \left\{ \begin{matrix} n + 99 & \text{~}\text{if}\text{~}n\text{~}\text{is odd}\text{~} \\ n - 98 & \text{~}\text{if}\text{~}n\text{~}\text{is even}\text{~} \end{matrix} \right.\ \end{matrix}$$

✔ (a) $a_{7 = 7 + 99 = 106}$

✔ (b) $a_{11} = 11 + 99 = 110$

✔ (c) $a_{8 = 8 - 98 = - 90}$

✘ (d) $a_{12} = 12 - 98 = - 86 \neq - 88$