Exam 13: Sequences and Series

Write $3.368368 \ldots$ as a fraction or improper fraction.

If $1-2+4-8+16-\cdots+256$ is a geometric series, give its ratio. If not, enter "not geometric"

Find the sum of the series: $11+\frac{11}{3}+\frac{11}{9}+\ldots$

If $6+9+12+15+18+21=\sum_{n=1}^{a}(b+c n)$ , then $a=\ldots, b=\ldots$ , and $c=$

A patient takes $600 \mathrm{mg}$ of a pain-killing drug twice a day. Every 12 hours, the patient's body metabolizes $30 \%$ of the drug present. How many mg of the drug remain in the patient's body after 8 days (right after the $16^{\text {th }}$ dose)? Round to 2 decimal places.

Find the sum $\sum_{i=3}^{\infty} \frac{4^{i}+2}{5^{i}}$ . Round your answer to 3 decimal places.

Does $\sum_{i=1}^{27}(i+5)^{2}=\sum_{j=6}^{32} j^{2}$ ?

An athlete is offered a signing bonus of $\$ 2.5$ million per year for 10 years. If the interest rate is $5.8 \%$ per year, compounded annually, what is the present value of this bonus? Round to the nearest dollar.

The following table shows the enrollment in a private college for the years 1996 to 2005. Interpret these figures as partial sums, $S_{n}$ , of terms of a sequence, $a_{n}$ , where $\mathrm{n}$ is the number of years since 1995 (so $S_{1}=4449, S_{2}=4478$ , etc.). year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 enrollment 4449 4478 4528 4575 4590 4562 4598 4693 4740 4773 Which of the following gives the total enrollment in 2004?

What is the $12^{\text {th }}$ term of the sequence $1,6,11,16, \ldots$ ? The sequence is either arithmetic or geometric.

Find the sum of the series $1-\frac{x}{6}+\frac{x^{2}}{36}-\frac{x^{3}}{216}+\cdots$ , for $|x| \leq 6$ .

Does the infinite geometric series $20-16+12.8-10.24+\cdots$ converge or diverge?

For the arithmetic sequence $a_{1}, a_{2}, \ldots, a_{n}$ , suppose $a_{1}=1, a_{2}=2$ , and the partial sum $S_{1}=1$ . What is $S_{6}$ ?

Does $\sum_{i=1}^{15} i^{2}-\sum_{j=1}^{16} j^{2}=-256$ ?

The following table shows the enrollment in a private college for the years 1996 to 2005. Interpret these figures as partial sums, $S_{n}$ , of terms of a sequence, $a_{n}$ , where $n$ is the number of years since 1995 (so $S_{1}=4194, S_{2}=4220$ , etc.). Which of the following gives the average enrollment between 1998 and 2001 ?

Find the sum of the first 400 odd integers.

What is $\sum_{k=1}^{30}(3 k+2) ?$

Find $\sum_{n=0}^{7} 6\left(\frac{1}{5}\right)^{n}$ to 3 decimal places.

Your doctor tells you to take $38 \mathrm{mg}$ of a certain drug every 12 hours. It is known that only $6 \%$ of the drug is still in the body at the end of 12 hours. What quantity of the drug is in the body right after taking the 11th dose? Give your answer correct to 3 decimal places.

A patient takes $400 \mathrm{mg}$ of a pain-killing drug twice a day. Every 12 hours, the patient's body metabolizes $25 \%$ of the drug present. If the patient stops taking the drug after 1 week, how many $\mathrm{mg}$ of the drug remain in the patient's body 2 days after that? Round to 2 decimal places.