For problems #1-9 write the explicit formula of a geometric sequence from the given information
1) The first term is \(11\) and the common ratio is \(4\)
2) \(a_1 = 5\) and \(r=\frac{1}{4}\)
3) \(a_3=4\) and \(r=2\)
4) \(a_1=9\) and \(a_4= 243\)
5) \(a_5=10\) and \(a_{13}=2560\)
6) \(a_2= 512\) and \(a_5=8\)
7) \(r=6\) and \(a_3=216\)
8) \(a_1=1\) and \(a_4=8\)
9) \(a_1=2\) and \(a_4=-54\)
For problems #10-12 determine if the sequence is geometric or not. If so, state the common ratio, if not explain why.
10) \(2, 4, 8, 15, 30, ...\)
11) \(16, 4, 1, \frac{1}{4}, ...\)
12) \(\frac{7}{3}, \frac{5}{3}, 1, \frac{1}{3}, ...\)
13) In a geometric sequence the common ratio is \(3\) and the first term is \(2\), Is \(9,565,938\) a term in the sequence? If so, what term is it?
14) In a geometric sequence the first term is \(8\) and the sixth term is \(-\frac{1}{4}\). Find the \(8\)th term.
15) In a geometric sequence if \(a_1=-5\) and \(r=-\frac{1}{3}\), find the sum of the first \(10\) terms. Round to the nearest hundredth.
16) Evaluate the sum \(\sum\limits_{i=1}^{\infty} 2(\frac{1}{4})^{i-1}\).
17) Evaluate the sum \(\sum\limits_{k=1}^{\infty} 5(2)^{k-1}\).
18) You are given a summer tutoring job. You agree to get paid \(\$200\) per week to begin with the understanding that you will get a \(5\%\) raise every week. If you work for \(10\) weeks this summer, how much money will you make altogether?
19) A pendulum swings through an arc measuring \(48\) inches on its first swing. On each subsequent swing it swings through \(80\%\) of its previous length. Approximately how many total inches does the arc swing through before it comes to a stop?
20) Explain thoroughly, in complete sentences, how to tell if an infinite geometric series converges or not. Provide specific examples to support your argument.
Review
21) What is the x-intercept of \(f(x)=\log_{2} (x-5)\)?
22) Solve for \(x\): \(4^{3x-1}=(\frac{1}{8}) ^x\).
23) Derive the following logarithmic property: \(\log_{b} a - \log_{b} c = \log_{b} (\frac{a}{c})\).
24) Simplify \(\ln(\log 10^e)\).
25) If you invest into a savings account that earns \(2.2\%\) annual interest compounded monthly, how long will it take to double your investment?
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