Exam 11: Further Topics in Algebra

Solve the problem. -Suppose that certain bacteria can double their size and divide every 30 minutes. Write a recursive sequence that describes this growth where each value of $\mathrm { n }$ represents a 30 -minute interval. Let $\mathrm { a } _ { 1 }$ $= 498$ represent the initial number of bacteria present.

Decide whether the given sequence is finite or infinite. -10, 11, 12, 13

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \left( \frac { 1 } { 30 } + n \right) ^ { 1 / n }$

Solve the problem. -A woman borrowed $10,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 5 years, how much did she pay?

Write out the first five terms of the sequence. - $a _ { n } = \frac { 6 n - 1 } { n ^ { 2 } + 6 n }$

Solve the problem. -A shirt company has 3 designs that can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company?

Solve the problem. -How many 5-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, if repetitions are not allowed?

Solve the problem. -A bag contains 6 apples and 4 oranges. If you select 5 pieces of fruit without looking, how many ways can you get 5 oranges?

Evaluate the sum using the given information. - $x _ { 1 } = 4 , x _ { 2 } = 3 , x _ { 3 } = - 1 , x _ { 4 } = - 4$ , and $\Delta x = 0.9 ; f ( x ) = x ^ { 2 } - 5$ $\sum _ { \mathrm { i } = 1 } ^ { 4 } \mathrm { f } \left( \mathrm { x } _ { \mathrm { i } } \right) \Delta \mathrm { x }$

Find a general term an for the geometric sequence. - $a _ { 1 } = - \frac { 1 } { 3 } , r = - 6$

Use mathematical induction to prove that the statement is true for every positive integer n. -If $0 < a < 1$ , then $a ^ { n } < a ^ { n - 1 }$ . (Assume that a is a constant.)

Evaluate the sum. - $\sum _ { \mathrm { j } = 100 } ^ { 5000 } \mathrm { j }$

Find the common ratio r for the given infinite geometric sequence. - $3 , - 9,27 , - 81,243 , \ldots$

Provide an appropriate response. -Consider the selections of seven apples from a barrel of 75 apples. Is this a combination, a permutation, or neither?

Solve the problem. -100 employees of a company are asked how they get to work and whether they work full time or part time. The figure below shows the results. If one of the 100 employees is randomly selected, Find the probability that the person does not commute by public transportation. 1. Public transportation: 8 full time, 9 part time 2) Bicycle: 4 full time, 4 part time 3) Drive alone: 27 full time, 33 part time 4) Car pool: 6 full time, 9 part time

In your own words, explain how to determine the binomial coefficient for $x ^ { 3 } y ^ { 5 }$ in the expansion of $( x + y ) ^ { 8 }$

What is true of both the first term and the last term in a binomial expansion?

Solve the problem. -A railroad bridge is built over a $V$ -shaped canyon. If the supports are 7 feet long at each end, the center support is 388 feet long, and the supports increase in length by 3 feet with each support (from 7 feet to 388 feet), what is the total length of all the supports?

Decide whether the given sequence is finite or infinite. - $10,11,12,13 , \ldots$

Write the indicated term of the binomial expansion. - $( x - 2 y ) ^ { 9 } ; 8$ th term