Module 2 Problem Set

Sequences and Their Notation

1. Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

2. Describe three ways that a sequence can be defined.

3. Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

4. What happens to the terms [latex]{a}_{n}[/latex] of a sequence when there is a negative factor in the formula that is raised to a power that includes [latex]n?[/latex] What is the term used to describe this phenomenon?

5. What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

For the following exercises, write the first four terms of the sequence.

6. [latex]{a}_{n}={2}^{n}-2[/latex]

7. [latex]{a}_{n}=-\frac{16}{n+1}[/latex]

8. [latex]{a}_{n}=-{\left(-5\right)}^{n - 1}[/latex]

9. [latex]{a}_{n}=\frac{{2}^{n}}{{n}^{3}}[/latex]

10. [latex]{a}_{n}=\frac{2n+1}{{n}^{3}}[/latex]

11. [latex]{a}_{n}=1.25\cdot {\left(-4\right)}^{n - 1}[/latex]

12. [latex]{a}_{n}=-4\cdot {\left(-6\right)}^{n - 1}[/latex]

13. [latex]{a}_{n}=\frac{{n}^{2}}{2n+1}[/latex]

14. [latex]{a}_{n}={\left(-10\right)}^{n}+1[/latex]

15. [latex]{a}_{n}=-\left(\frac{4\cdot {\left(-5\right)}^{n - 1}}{5}\right)[/latex]

For the following exercises, write the first five terms of the sequence.

16. [latex]{a}_{1}=9,\text{ }{a}_{n}={a}_{n - 1}+n[/latex]

17. [latex]{a}_{1}=3,\text{ }{a}_{n}=\left(-3\right){a}_{n - 1}[/latex]

18. [latex]{a}_{1}=-4,\text{ }{a}_{n}=\frac{{a}_{n - 1}+2n}{{a}_{n - 1}-1}[/latex]

19. [latex]{a}_{1}=-1,\text{ }{a}_{n}=\frac{{\left(-3\right)}^{n - 1}}{{a}_{n - 1}-2}[/latex]

20. [latex]{a}_{1}=-30,\text{ }{a}_{n}=\left(2+{a}_{n - 1}\right){\left(\frac{1}{2}\right)}^{n}[/latex]

For the following exercises, write the first eight terms of the sequence.

21. [latex]{a}_{1}=\frac{1}{24},{\text{ a}}_{2}=1,\text{ }{a}_{n}=\left(2{a}_{n - 2}\right)\left(3{a}_{n - 1}\right)[/latex]

22. [latex]{a}_{1}=-1,{\text{ a}}_{2}=5,\text{ }{a}_{n}={a}_{n - 2}\left(3-{a}_{n - 1}\right)[/latex]

23. [latex]{a}_{1}=2,{\text{ a}}_{2}=10,\text{ }{a}_{n}=\frac{2\left({a}_{n - 1}+2\right)}{{a}_{n - 2}}[/latex]

For the following exercises, write a explicit formula for each sequence.

24. [latex]-2.5,-5,-10,-20,-40,\dots[/latex]

25. [latex]-8,-6,-3,1,6,\dots[/latex]

26. [latex]2,\text{ }4,\text{ }12,\text{ }48,\text{ }240,\text{ }\dots[/latex]

27. [latex]35,\text{ }38,\text{ }41,\text{ }44,\text{ }47,\text{ }\dots[/latex]

28. [latex]15,3,\frac{3}{5},\frac{3}{25},\frac{3}{125},\cdots[/latex]

For the following exercises, evaluate the factorial.

29. [latex]6![/latex]

30. [latex]\left(\frac{12}{6}\right)![/latex]

31. [latex]\frac{12!}{6!}[/latex]

32. [latex]\frac{100!}{99!}[/latex]

For the following exercises, write the first four terms of the sequence.

33. [latex]{a}_{n}=\frac{n!}{{n}^{\text{2}}}[/latex]

34. [latex]{a}_{n}=\frac{3\cdot n!}{4\cdot n!}[/latex]

35. [latex]{a}_{n}=\frac{n!}{{n}^{2}-n - 1}[/latex]

36. [latex]{a}_{n}=\frac{100\cdot n}{n\left(n - 1\right)!}[/latex]

Arithmetic Sequences

1. What is an arithmetic sequence?

2. How is the common difference of an arithmetic sequence found?

3. How do we determine whether a sequence is arithmetic?

4. Give an example of an arithmetic sequence.

5. Give an example of a sequence that is not arithmetic.

For the following exercises, find the common difference for the arithmetic sequence provided.

6. [latex]\left\{5,11,17,23,29,...\right\}[/latex]

7. [latex]\left\{0,\frac{1}{2},1,\frac{3}{2},2,...\right\}[/latex]

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

8. [latex]\left\{11.4,9.3,7.2,5.1,3,...\right\}[/latex]

9. [latex]\left\{4,16,64,256,1024,...\right\}[/latex]

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.

10. [latex]{a}_{1}=-25[/latex] , [latex]d=-9[/latex]

11. [latex]{a}_{1}=0[/latex] , [latex]d=\frac{2}{3}[/latex]

For the following exercises, write the first five terms of the arithmetic series given two terms.

12. [latex]{a}_{1}=17,{a}_{7}=-31[/latex]

13. [latex]{a}_{13}=-60,{a}_{33}=-160[/latex]

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

14. First term is 3, common difference is 4, find the 5th term.

15. First term is 4, common difference is 5, find the 4th term.

16. First term is 5, common difference is 6, find the 8th term.

17. First term is 6, common difference is 7, find the 6th term.

18. First term is 7, common difference is 8, find the 7th term.

For the following exercises, find the first term given two terms from an arithmetic sequence.

19. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{6}=12[/latex] and [latex]{a}_{14}=28[/latex].

20. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{7}=21[/latex] and [latex]{a}_{15}=42[/latex].

21. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{8}=40[/latex] and [latex]{a}_{23}=115[/latex].

22. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{9}=54[/latex] and [latex]{a}_{17}=102[/latex].

23. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{11}=11[/latex] and [latex]{a}_{21}=16[/latex].

For the following exercises, find the specified term given two terms from an arithmetic sequence.

24. [latex]{a}_{1}=33[/latex] and [latex]{a}_{7}=-15[/latex]. Find [latex]{a}_{4}[/latex].

25. [latex]{a}_{3}=-17.1[/latex] and [latex]{a}_{10}=-15.7[/latex]. Find [latex]{a}_{21}[/latex].

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.

26. [latex]{a}_{n}=24 - 4n[/latex]

27. [latex]{a}_{n}=\frac{1}{2}n-\frac{1}{2}[/latex]

For the following exercises, write an explicit formula for each arithmetic sequence.

28. [latex]{a}_{n}=\left\{3,5,7,...\right\}[/latex]

29. [latex]{a}_{n}=\left\{32,24,16,...\right\}[/latex]

30. [latex]{a}_{n}=\left\{-5\text{, }95\text{, }195\text{, }...\right\}[/latex]

31. [latex]{a}_{n}=\left\{-17\text{, }-217\text{, }-417\text{,}...\right\}[/latex]

32. [latex]{a}_{n}=\left\{1.8\text{, }3.6\text{, }5.4\text{, }...\right\}[/latex]

33. [latex]{a}_{n}=\left\{15.8,18.5,21.2,...\right\}[/latex]

34. [latex]{a}_{n}=\left\{\frac{1}{3},-\frac{4}{3},-3\text{, }...\right\}[/latex]

35. [latex]{a}_{n}=\left\{0,\frac{1}{3},\frac{2}{3},...\right\}[/latex]

36. [latex]{a}_{n}=\left\{-5,-\frac{10}{3},-\frac{5}{3},\dots \right\}[/latex]

For the following exercises, find the number of terms in the given finite arithmetic sequence.

37. [latex]{a}_{n}=\left\{3\text{,}-4\text{,}-11\text{, }...\text{,}-60\right\}[/latex]

38. [latex]{a}_{n}=\left\{1.2,1.4,1.6,...,3.8\right\}[/latex]

39. [latex]{a}_{n}=\left\{\frac{1}{2},2,\frac{7}{2},...,8\right\}[/latex]

Geometric Sequences

1. What is a geometric sequence?

2. How is the common ratio of a geometric sequence found?

3. What is the procedure for determining whether a sequence is geometric?

4. What is the difference between an arithmetic sequence and a geometric sequence?

5. Describe how exponential functions and geometric sequences are similar. How are they different?

For the following exercises, find the common ratio for the geometric sequence.

6. [latex]1,3,9,27,81,..[/latex].

7. [latex]-0.125,0.25,-0.5,1,-2,..[/latex].

8. [latex]-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128},..[/latex].

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

9. [latex]-6,-12,-24,-48,-96,..[/latex].

10. [latex]5,5.2,5.4,5.6,5.8,..[/latex].

11. [latex]-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},..[/latex].

12. [latex]6,8,11,15,20,..[/latex].

13. [latex]0.8,4,20,100,500,..[/latex].

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.

14. [latex]\begin{array}{cc}{a}_{1}=8,& r=0.3\end{array}[/latex]

15. [latex]\begin{array}{cc}{a}_{1}=5,& r=\frac{1}{5}\end{array}[/latex]

For the following exercises, write the first five terms of the geometric sequence, given any two terms.

16. [latex]\begin{array}{cc}{a}_{7}=64,& {a}_{10}\end{array}=512[/latex]

17. [latex]\begin{array}{cc}{a}_{6}=25,& {a}_{8}\end{array}=6.25[/latex]

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.

18. The first term is [latex]2[/latex], and the common ratio is [latex]3[/latex]. Find the 5th term.

19. The first term is 16 and the common ratio is [latex]-\frac{1}{3}[/latex]. Find the 4th term.

For the following exercises, find the specified term for the geometric sequence, given the first four terms.

20. [latex]{a}_{n}=\left\{-1,2,-4,8,...\right\}[/latex]. Find [latex]{a}_{12}[/latex].

21. [latex]{a}_{n}=\left\{-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},...\right\}[/latex]. Find [latex]{a}_{7}[/latex].

For the following exercises, write the first five terms of the geometric sequence.

22. [latex]\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\frac{1}{3}\end{array}{a}_{n - 1}[/latex]

23. [latex]\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\end{array}[/latex]

For the following exercises, write an explicit formula for each geometric sequence.

24. [latex]{a}_{n}=\left\{-1,5,-25,125,...\right\}[/latex]

25. [latex]{a}_{n}=\left\{-32,-16,-8,-4,...\right\}[/latex]

26. [latex]{a}_{n}=\left\{14,56,224,896,...\right\}[/latex]

27. [latex]{a}_{n}=\left\{10,-3,0.9,-0.27,...\right\}[/latex]

28. [latex]{a}_{n}=\left\{0.61,1.83,5.49,16.47,...\right\}[/latex]

29. [latex]{a}_{n}=\left\{\frac{3}{5},\frac{1}{10},\frac{1}{60},\frac{1}{360},...\right\}[/latex]

30. [latex]{a}_{n}=\left\{-2,\frac{4}{3},-\frac{8}{9},\frac{16}{27},...\right\}[/latex]

31. [latex]{a}_{n}=\left\{\frac{1}{512},-\frac{1}{128},\frac{1}{32},-\frac{1}{8},...\right\}[/latex]

For the following exercises, write the first five terms of the geometric sequence.

32. [latex]{a}_{n}=-4\cdot {5}^{n - 1}[/latex]

33. [latex]{a}_{n}=12\cdot {\left(-\frac{1}{2}\right)}^{n - 1}[/latex]

For the following exercises, find the number of terms in the given finite geometric sequence.

34. [latex]{a}_{n}=\left\{-1,3,-9,...,2187\right\}[/latex]

35. [latex]{a}_{n}=\left\{2,1,\frac{1}{2},...,\frac{1}{1024}\right\}[/latex]

Fractals

Using the initiator and generator shown, draw the next two stages of the iterated fractal.

1.

Initiator is a horizontal line. Generator is a horizontal line that then goes up at a right angle, right at a right angle, down at a right angle, and then continues horizontally.

 2.

Initiator is a horizontal line. Generator is a zig-zag.
3.

Initiator is an upward-sloping line. Generator is that line with smaller lines branching off of it.

 4.

Initiator is a horizontal line. Generator is two short horizontal lines side-by-side.
5.

Initiator is a square. Generator is eight more squares arranged to form the border of a large square.

 6.

Initiator is an equilateral triangle. Generator is three equilateral triangles that touch each other at an angle.
  1. Create your own version of Sierpinski gasket with added randomness.
  2. Create a version of the branching tree fractal from example #3 with added randomness.
  3. Determine the fractal dimension of the Koch curve.
  4. Determine the fractal dimension of the curve generated in exercise #1
  5. Determine the fractal dimension of the Sierpinski carpet generated in exercise #5
  6. Determine the fractal dimension of the Cantor set generated in exercise #4