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B-9.

Consider the sequence that begins $40,20,10,5,\dots$

1. Based on the information given, can this sequence be arithmetic? Can it be geometric? Why?

In an arithmetic sequence the difference between sequential terms is constant. Each term of an arithmetic sequence can be generated by adding the common difference to the previous term. For example in the sequence, $4$, $7$, $10$, $13$, …, the common difference is $3$.

A geometric sequence is a sequence that is generated by a multiplier. This means that each term of a geometric sequence can be found by multiplying the previous term by a constant. For example: $5$, $15$, $45$… is the beginning of a geometric sequence with generator (common ratio) $3$. In general a geometric sequence can be represented $a$, $ar$, $ar^{2}$, ... $+ ar^{n-1}$.

2. Assume this is a geometric sequence. On graph paper, plot the sequence on a graph up to $n=6$.

Start by making a table of values.

3. Will the values of the sequence ever become zero or negative? Explain.

No. Multiplying by $0.5$ will never yield $0$ unless the other factor is $0$, which is impossible in this sequence.
When multiplying two positive numbers, a negative number can never be obtained.