Geometric Sequences

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Problems about geometric (and other) sequences.

A sequence has first two terms $1, 2, \dots$ . What type of sequence is this?

An arithmetic sequence. A geometric sequence. A quadratic sequence. It is impossible to tell.

If we do not know the rule for generating the terms of this sequence, can we be sure we know the next term?

Sylvie has a bank account which contained $\$400$ back when interest rates were high. When she graduated from high school, she decided to invest the money in a certificate of deposit (CD) that yielded $10\%$ interest, compounded annually.

Would we use a geometric sequence to describe the value of her CD?

Yes, this is a geometric sequence. No, this is not a geometric sequence.

Fill out the following table corresponding to the value of her CD $n$ years later. (Enter the exact amount; do not round to the nearest cent.) $\begin {array}{c|c} \hline n & \text {Value} \\ \hline 0 & \answer {400}\\ 1 & \answer {440} \\ 2 & \answer {484} \\ 3 & \answer {532.4}\\ 4 & \answer {585.64}\\ 5 & \answer {644.204}\\ 6 & \answer {708.6244}\\ 7 & \answer {779.48684}\\ \end {array}$

Write a recursive formula for the value of Sylvie’s account in year $n$ , using “Next” and “Now” to describe the situation.

Sylvie account value: $=$ $\times$ Next Now $n$ 400 1.1 0.1 .

Write an explicit formula for $f(n)$ , the value of her CD in year $n$ .

Sylvie CD will be valued at $f(n) = \answer [given]{400(1.1)^n}$ in $n$ years.

Seth is in a lab, measuring the amount of a decaying substance. He knows that each day, he expects to have $\frac {11}{12}$ of the amount he had the previous day. If he began with $90$ grams of the substance, predict how many grams he will have in the future.

How much of the substance will Seth expect to have on day $5$ ? Seth expects to have $\answer [given]{90(11/12)^{5}}$ grams of the substance.

When will Seth expect to have $75$ grams of the substance? Seth expects to have $75$ grams of the substance sometime between day $\answer [given]{2}$ and day $\answer [given]{3}$ .

Write a recursive formula for the amount of the substance Seth expects to have on day $n$ . Use $f(n-1)$ in your equation. Seth expects to have $f(n) = \answer [given]{f(n-1)(11/12)}$ grams of the substance.

Write an explicit formula for the amount of the substance Seth expects to have on day $n$ . Seth expects to have $f(n) = \answer [given]{90(11/12)^n}$ grams of the substance.

Assume the sequence below is a geometric sequence. Fill in the blanks.

$\answer {3}, 6, 12, \answer {24}, \answer {48}, \answer {96}, \answer {192}, \dots$

Assume the sequence below is a geometric sequence. Fill in the blanks.

$\dots , \answer {-4}, 6, \answer {-9}, \answer {13.5}, -20.25, \answer {30.375}, \answer {-45.5625}, \dots$