A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 16 , a _ { 2 } = 8$ . Use your calculator to guess the limit of the sequence.
A) 13
B) $6$
C) 12
D) $17$
E) $26$

Question

Quizplus · Accepted Answer

By repeatedly applying the recursive formula, we can generate the following terms of the sequence: 16, 8, 12, 10, 11, 10.5, 10.75, 10.625, 10.6875, 10.65625, 10.671875, ... It appears that the terms are approaching a limit of about 12, so (C) is the best choice. We could also verify this by computing more terms and observing that they get progressively closer to 12.

A Sequenceis {an}\left\{ a _ { n } \right\}{an​} Defined Recursively by the Equation

A Sequenceis $\left\{ a _ { n } \right\}$ Defined Recursively by the Equation