91Ó°ÊÓ

Perform the indicated operation and, if possible, simplify. $$ \frac{x^{2}-6 x+8}{4 x+12} \cdot \frac{x+3}{x^{2}-4} $$

Perform the indicated operation and, if possible, simplify. $$ \frac{y^{3}-y}{3 y+1} \div \frac{y^{2}}{9 y+3} $$

Explain why the equation $$ \sum_{k=1}^{n}\left(a_{k}+b_{k}\right)=\sum_{k=1}^{n} a_{k}+\sum_{k=1}^{n} b_{k} $$ is true for any positive integer \(n .\) What laws are used to justify this result?

The sides of a square are each \(16 \mathrm{cm}\) long. A second square is inscribed by joining the midpoints of the sides, successively. In the second square we repeat the process, inscribing a third square. If this process is continued indefinitely, what is the sum of all of the areas of all the squares? (Hint: Use an infinite geometric series.)

Is it true that $$ \sum_{k=1}^{n} c a_{k}=c \sum_{k=1}^{n} a_{k} ? $$ Why or why not?

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=1, a_{n+1}=5 a_{n}-2 $$

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=0, a_{n+1}=\left(a_{n}\right)^{2}+3 $$

The value of an LCD projector is \(\$ 2500 .\) Its resale value each year is \(80 \%\) of its value the year before. Write a sequence listing the resale value of the machine at the start of each year for a 10-year period.

A single cell of bacterium divides into two every 15 min. Suppose that the same rate of division is maintained for 4 hr. Write a sequence listing the number of cells after successive \(15-\min\) periods.

Find the first five terms of each sequence; then find \(S_{5}\). $$ a_{n}=\frac{1}{2^{n}} \log 1000^{n} $$