Answer :
Alright, let's tackle this exercise step-by-step.
### Part (a) Count in thirds from [tex]\(\frac{1}{3}\)[/tex] to [tex]\(3 \frac{2}{3}\)[/tex]:
We start at [tex]\(\frac{1}{3}\)[/tex] and add [tex]\(\frac{1}{3}\)[/tex] incrementally until we reach [tex]\(3 \frac{2}{3}\)[/tex].
1. [tex]\(\frac{1}{3}\)[/tex]
2. [tex]\(\frac{1}{3} + \frac{1}{3} = \frac{2}{3}\)[/tex]
3. [tex]\(\frac{2}{3} + \frac{1}{3} = 1\)[/tex]
4. [tex]\(1 + \frac{1}{3} = 1 \frac{1}{3}\)[/tex]
5. [tex]\(1 \frac{1}{3} + \frac{1}{3} = 1 \frac{2}{3}\)[/tex]
6. [tex]\(1 \frac{2}{3} + \frac{1}{3} = 2\)[/tex]
7. [tex]\(2 + \frac{1}{3} = 2 \frac{1}{3}\)[/tex]
8. [tex]\(2 \frac{1}{3} + \frac{1}{3} = 2 \frac{2}{3}\)[/tex]
9. [tex]\(2 \frac{2}{3} + \frac{1}{3} = 3\)[/tex]
10. [tex]\(3 + \frac{1}{3} = 3 \frac{1}{3}\)[/tex]
11. [tex]\(3 \frac{1}{3} + \frac{1}{3} = 3 \frac{2}{3}\)[/tex]
So the sequence in decimals is:
[tex]\[ [0.3333333333333333, 0.6666666666666666, 1.0, 1.3333333333333333, 1.6666666666666665, 1.9999999999999998, 2.333333333333333, 2.6666666666666665, 3.0, 3.3333333333333335 ] \][/tex]
### Part (b) Count in quarters from [tex]\(\frac{1}{4}\)[/tex] to [tex]\(4 \frac{3}{4}\)[/tex]:
We start at [tex]\(\frac{1}{4}\)[/tex] and add [tex]\(\frac{1}{4}\)[/tex] incrementally until we reach [tex]\(4 \frac{3}{4}\)[/tex].
1. [tex]\(\frac{1}{4}\)[/tex]
2. [tex]\(\frac{1}{4} + \(\frac{1}{4}\)[/tex] = [tex]\(\frac{1}{2}\)[/tex]\)
3. [tex]\(\(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = [tex]\(\frac{3}{4}\)[/tex]\)
4. [tex]\(\(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 1\)
5. [tex]\(1 + \(\frac{1}{4}\)[/tex] = 1 [tex]\(\frac{1}{4}\)[/tex]\)
6. [tex]\(1 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 1 [tex]\(\frac{1}{2}\)[/tex]\)
7. [tex]\(1 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 1 [tex]\(\frac{3}{4}\)[/tex]\)
8. [tex]\(1 \(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 2\)
9. [tex]\(2 + \(\frac{1}{4}\)[/tex] = 2 [tex]\(\frac{1}{4}\)[/tex]\)
10. [tex]\(2 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 2 [tex]\(\frac{1}{2}\)[/tex]\)
11. [tex]\(2 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 2 [tex]\(\frac{3}{4}\)[/tex]\)
12. [tex]\(2 \(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 3\)
13. [tex]\(3 + \(\frac{1}{4}\)[/tex] = 3 [tex]\(\frac{1}{4}\)[/tex]\)
14. [tex]\(3 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 3 [tex]\(\frac{1}{2}\)[/tex]\)
15. [tex]\(3 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 3 [tex]\(\frac{3}{4}\)[/tex]\)
16. [tex]\(3 \(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 4\)
17. [tex]\(4 + \(\frac{1}{4}\)[/tex] = 4 [tex]\(\frac{1}{4}\)[/tex]\)
18. [tex]\(4 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 4 [tex]\(\frac{1}{2}\)[/tex]\)
19. [tex]\(4 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 4 [tex]\(\frac{3}{4}\)[/tex]\)
So the sequence in decimals is:
[tex]\[ [0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0, 4.25, 4.5, 4.75] \][/tex]
### Part (c) Count in fifths from [tex]\(\frac{2}{5}\)[/tex] to [tex]\(3 \frac{3}{5}\)[/tex]:
We start at [tex]\(\frac{2}{5}\)[/tex] and add [tex]\(\frac{1}{5}\)[/tex] incrementally until we reach [tex]\(3 \frac{3}{5}\)[/tex].
1. [tex]\(\frac{2}{5}\)[/tex]
2. [tex]\(\frac{2}{5} + \(\frac{1}{5}\)[/tex] = [tex]\(\frac{3}{5}\)[/tex]\)
3. [tex]\(\(\frac{3}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = [tex]\(\frac{4}{5}\)[/tex]\)
4. [tex]\(\(\frac{4}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1\)
5. [tex]\(1 + \(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{1}{5}\)[/tex]\)
6. [tex]\(1 \(\frac{1}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{2}{5}\)[/tex]\)
7. [tex]\(1 \(\frac{2}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{3}{5}\)[/tex]\)
8. [tex]\(1 \(\frac{3}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{4}{5}\)[/tex]\)
9. [tex]\(1 \(\frac{4}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2\)
10. [tex]\(2 + \(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{1}{5}\)[/tex]\)
11. [tex]\(2 \(\frac{1}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{2}{5}\)[/tex]\)
12. [tex]\(2 \(\frac{2}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{3}{5}\)[/tex]\)
13. [tex]\(2 \(\frac{3}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{4}{5}\)[/tex]\)
14. [tex]\(2 \(\frac{4}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 3\)
15. [tex]\(3 + \(\frac{1}{5}\)[/tex] = 3 [tex]\(\frac{1}{5}\)[/tex]\)
16. [tex]\(3 \(\frac{1}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 3 [tex]\(\frac{2}{5}\)[/tex]\)
17. [tex]\(3 \(\frac{2}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 3 [tex]\(\frac{3}{5}\)[/tex]\)
So the sequence in decimals is:
[tex]\[ [0.4, 0.6000000000000001, 0.8, 1.0, 1.2, 1.4, 1.5999999999999999, 1.7999999999999998, 1.9999999999999998, 2.1999999999999997, 2.4, 2.6, 2.8000000000000003, 3.0000000000000004, 3.2000000000000006, 3.400000000000001] \][/tex]
### Part (a) Count in thirds from [tex]\(\frac{1}{3}\)[/tex] to [tex]\(3 \frac{2}{3}\)[/tex]:
We start at [tex]\(\frac{1}{3}\)[/tex] and add [tex]\(\frac{1}{3}\)[/tex] incrementally until we reach [tex]\(3 \frac{2}{3}\)[/tex].
1. [tex]\(\frac{1}{3}\)[/tex]
2. [tex]\(\frac{1}{3} + \frac{1}{3} = \frac{2}{3}\)[/tex]
3. [tex]\(\frac{2}{3} + \frac{1}{3} = 1\)[/tex]
4. [tex]\(1 + \frac{1}{3} = 1 \frac{1}{3}\)[/tex]
5. [tex]\(1 \frac{1}{3} + \frac{1}{3} = 1 \frac{2}{3}\)[/tex]
6. [tex]\(1 \frac{2}{3} + \frac{1}{3} = 2\)[/tex]
7. [tex]\(2 + \frac{1}{3} = 2 \frac{1}{3}\)[/tex]
8. [tex]\(2 \frac{1}{3} + \frac{1}{3} = 2 \frac{2}{3}\)[/tex]
9. [tex]\(2 \frac{2}{3} + \frac{1}{3} = 3\)[/tex]
10. [tex]\(3 + \frac{1}{3} = 3 \frac{1}{3}\)[/tex]
11. [tex]\(3 \frac{1}{3} + \frac{1}{3} = 3 \frac{2}{3}\)[/tex]
So the sequence in decimals is:
[tex]\[ [0.3333333333333333, 0.6666666666666666, 1.0, 1.3333333333333333, 1.6666666666666665, 1.9999999999999998, 2.333333333333333, 2.6666666666666665, 3.0, 3.3333333333333335 ] \][/tex]
### Part (b) Count in quarters from [tex]\(\frac{1}{4}\)[/tex] to [tex]\(4 \frac{3}{4}\)[/tex]:
We start at [tex]\(\frac{1}{4}\)[/tex] and add [tex]\(\frac{1}{4}\)[/tex] incrementally until we reach [tex]\(4 \frac{3}{4}\)[/tex].
1. [tex]\(\frac{1}{4}\)[/tex]
2. [tex]\(\frac{1}{4} + \(\frac{1}{4}\)[/tex] = [tex]\(\frac{1}{2}\)[/tex]\)
3. [tex]\(\(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = [tex]\(\frac{3}{4}\)[/tex]\)
4. [tex]\(\(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 1\)
5. [tex]\(1 + \(\frac{1}{4}\)[/tex] = 1 [tex]\(\frac{1}{4}\)[/tex]\)
6. [tex]\(1 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 1 [tex]\(\frac{1}{2}\)[/tex]\)
7. [tex]\(1 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 1 [tex]\(\frac{3}{4}\)[/tex]\)
8. [tex]\(1 \(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 2\)
9. [tex]\(2 + \(\frac{1}{4}\)[/tex] = 2 [tex]\(\frac{1}{4}\)[/tex]\)
10. [tex]\(2 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 2 [tex]\(\frac{1}{2}\)[/tex]\)
11. [tex]\(2 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 2 [tex]\(\frac{3}{4}\)[/tex]\)
12. [tex]\(2 \(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 3\)
13. [tex]\(3 + \(\frac{1}{4}\)[/tex] = 3 [tex]\(\frac{1}{4}\)[/tex]\)
14. [tex]\(3 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 3 [tex]\(\frac{1}{2}\)[/tex]\)
15. [tex]\(3 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 3 [tex]\(\frac{3}{4}\)[/tex]\)
16. [tex]\(3 \(\frac{3}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 4\)
17. [tex]\(4 + \(\frac{1}{4}\)[/tex] = 4 [tex]\(\frac{1}{4}\)[/tex]\)
18. [tex]\(4 \(\frac{1}{4}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 4 [tex]\(\frac{1}{2}\)[/tex]\)
19. [tex]\(4 \(\frac{1}{2}\)[/tex] + [tex]\(\frac{1}{4}\)[/tex] = 4 [tex]\(\frac{3}{4}\)[/tex]\)
So the sequence in decimals is:
[tex]\[ [0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0, 4.25, 4.5, 4.75] \][/tex]
### Part (c) Count in fifths from [tex]\(\frac{2}{5}\)[/tex] to [tex]\(3 \frac{3}{5}\)[/tex]:
We start at [tex]\(\frac{2}{5}\)[/tex] and add [tex]\(\frac{1}{5}\)[/tex] incrementally until we reach [tex]\(3 \frac{3}{5}\)[/tex].
1. [tex]\(\frac{2}{5}\)[/tex]
2. [tex]\(\frac{2}{5} + \(\frac{1}{5}\)[/tex] = [tex]\(\frac{3}{5}\)[/tex]\)
3. [tex]\(\(\frac{3}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = [tex]\(\frac{4}{5}\)[/tex]\)
4. [tex]\(\(\frac{4}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1\)
5. [tex]\(1 + \(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{1}{5}\)[/tex]\)
6. [tex]\(1 \(\frac{1}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{2}{5}\)[/tex]\)
7. [tex]\(1 \(\frac{2}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{3}{5}\)[/tex]\)
8. [tex]\(1 \(\frac{3}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 1 [tex]\(\frac{4}{5}\)[/tex]\)
9. [tex]\(1 \(\frac{4}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2\)
10. [tex]\(2 + \(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{1}{5}\)[/tex]\)
11. [tex]\(2 \(\frac{1}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{2}{5}\)[/tex]\)
12. [tex]\(2 \(\frac{2}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{3}{5}\)[/tex]\)
13. [tex]\(2 \(\frac{3}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 2 [tex]\(\frac{4}{5}\)[/tex]\)
14. [tex]\(2 \(\frac{4}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 3\)
15. [tex]\(3 + \(\frac{1}{5}\)[/tex] = 3 [tex]\(\frac{1}{5}\)[/tex]\)
16. [tex]\(3 \(\frac{1}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 3 [tex]\(\frac{2}{5}\)[/tex]\)
17. [tex]\(3 \(\frac{2}{5}\)[/tex] + [tex]\(\frac{1}{5}\)[/tex] = 3 [tex]\(\frac{3}{5}\)[/tex]\)
So the sequence in decimals is:
[tex]\[ [0.4, 0.6000000000000001, 0.8, 1.0, 1.2, 1.4, 1.5999999999999999, 1.7999999999999998, 1.9999999999999998, 2.1999999999999997, 2.4, 2.6, 2.8000000000000003, 3.0000000000000004, 3.2000000000000006, 3.400000000000001] \][/tex]
