Work through these Fibonacci sequence problems with help from the accompanying video tutorial and QR code for instant access.

Problem Analysis

The image contains questions related to Fibonacci sequences and similar patterns. Let's break down the problem step by step:

#### Question 1:
The first 4 numbers in the Fibonacci sequence are given as \(1, 1, 2, 3\).

1. (a) What is the 5th term of the Fibonacci sequence?
2. (b) What is the 6th term of the Fibonacci sequence?
3. (c) Describe the rule for continuing the Fibonacci sequence.

#### Question 2:
Find the next three terms of the following Fibonacci-style sequences:
- (a) \(2, 4, 6, 10, \ldots\)
- (b) \(3, 6, 9, 15, \ldots\)
- (c) \(4, 8, 12, 20, \ldots\)
- (d) \(15, 23, 38, 62, \ldots\)
- (e) \(5, 12, 17, 29, \ldots\)
- (f) \(-3, 5, 2, 7, \ldots\)
- (g) \(35, 60, 95, 155, \ldots\)
- (h) \(-1, -3, -4, -7, \ldots\)
- (i) \(1.2, 2.7, 3.9, 6.6, \ldots\)
- (j) \(0.11, 2.32, 2.43, 4.75, \ldots\)
- (k) \(-5.1, 1.1, -4, -2.9, \ldots\)
- (l) \(-0.5, -0.7, -1.2, -1.9, \ldots\)
- (m) \(\frac{1}{11}, \frac{3}{11}, \frac{4}{11}, \frac{7}{11}, \ldots\)
- (n) \(\frac{5}{6}, \frac{11}{12}, \frac{7}{4}, \frac{8}{3}, \ldots\)
- (o) \(-\frac{1}{5}, \frac{1}{2}, \frac{3}{10}, \frac{4}{5}, \ldots\)

---

Solution

#### Question 1:
The Fibonacci sequence is defined such that each term is the sum of the two preceding terms. The general rule is:
\[
F_n = F_{n-1} + F_{n-2}
\]
where \(F_1 = 1\) and \(F_2 = 1\).

1. (a) What is the 5th term of the Fibonacci sequence?
- Given: \(F_1 = 1\), \(F_2 = 1\), \(F_3 = 2\), \(F_4 = 3\).
- Calculate \(F_5\):
\[
F_5 = F_4 + F_3 = 3 + 2 = 5
\]
- Answer: \(F_5 = 5\).

2. (b) What is the 6th term of the Fibonacci sequence?
- Calculate \(F_6\):
\[
F_6 = F_5 + F_4 = 5 + 3 = 8
\]
- Answer: \(F_6 = 8\).

3. (c) Describe the rule for continuing the Fibonacci sequence.
- The rule for continuing the Fibonacci sequence is:
\[
F_n = F_{n-1} + F_{n-2}
\]
where \(F_1 = 1\) and \(F_2 = 1\).

#### Question 2:
For each sequence, we need to identify the pattern and find the next three terms.

1. (a) \(2, 4, 6, 10, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
6 = 4 + 2, \quad 10 = 6 + 4
\]
- Next terms:
\[
10 + 6 = 16, \quad 16 + 10 = 26, \quad 26 + 16 = 42
\]
- Answer: \(16, 26, 42\).

2. (b) \(3, 6, 9, 15, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
9 = 6 + 3, \quad 15 = 9 + 6
\]
- Next terms:
\[
15 + 9 = 24, \quad 24 + 15 = 39, \quad 39 + 24 = 63
\]
- Answer: \(24, 39, 63\).

3. (c) \(4, 8, 12, 20, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
12 = 8 + 4, \quad 20 = 12 + 8
\]
- Next terms:
\[
20 + 12 = 32, \quad 32 + 20 = 52, \quad 52 + 32 = 84
\]
- Answer: \(32, 52, 84\).

4. (d) \(15, 23, 38, 62, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
38 = 23 + 15, \quad 62 = 38 + 23
\]
- Next terms:
\[
62 + 38 = 100, \quad 100 + 62 = 162, \quad 162 + 100 = 262
\]
- Answer: \(100, 162, 262\).

5. (e) \(5, 12, 17, 29, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
17 = 12 + 5, \quad 29 = 17 + 12
\]
- Next terms:
\[
29 + 17 = 46, \quad 46 + 29 = 75, \quad 75 + 46 = 121
\]
- Answer: \(46, 75, 121\).

6. (f) \(-3, 5, 2, 7, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
2 = -3 + 5, \quad 7 = 2 + 5
\]
- Next terms:
\[
7 + 2 = 9, \quad 9 + 7 = 16, \quad 16 + 9 = 25
\]
- Answer: \(9, 16, 25\).

7. (g) \(35, 60, 95, 155, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
95 = 60 + 35, \quad 155 = 95 + 60
\]
- Next terms:
\[
155 + 95 = 250, \quad 250 + 155 = 405, \quad 405 + 250 = 655
\]
- Answer: \(250, 405, 655\).

8. (h) \(-1, -3, -4, -7, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
-4 = -3 + (-1), \quad -7 = -4 + (-3)
\]
- Next terms:
\[
-7 + (-4) = -11, \quad -11 + (-7) = -18, \quad -18 + (-11) = -29
\]
- Answer: \(-11, -18, -29\).

9. (i) \(1.2, 2.7, 3.9, 6.6, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
3.9 = 2.7 + 1.2, \quad 6.6 = 3.9 + 2.7
\]
- Next terms:
\[
6.6 + 3.9 = 10.5, \quad 10.5 + 6.6 = 17.1, \quad 17.1 + 10.5 = 27.6
\]
- Answer: \(10.5, 17.1, 27.6\).

10. (j) \(0.11, 2.32, 2.43, 4.75, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
2.43 = 2.32 + 0.11, \quad 4.75 = 2.43 + 2.32
\]
- Next terms:
\[
4.75 + 2.43 = 7.18, \quad 7.18 + 4.75 = 11.93, \quad 11.93 + 7.18 = 19.11
\]
- Answer: \(7.18, 11.93, 19.11\).

11. (k) \(-5.1, 1.1, -4, -2.9, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
-4 = 1.1 + (-5.1), \quad -2.9 = -4 + 1.1
\]
- Next terms:
\[
-2.9 + (-4) = -6.9, \quad -6.9 + (-2.9) = -9.8, \quad -9.8 + (-6.9) = -16.7
\]
- Answer: \(-6.9, -9.8, -16.7\).

12. (l) \(-0.5, -0.7, -1.2, -1.9, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
-1.2 = -0.7 + (-0.5), \quad -1.9 = -1.2 + (-0.7)
\]
- Next terms:
\[
-1.9 + (-1.2) = -3.1, \quad -3.1 + (-1.9) = -5.0, \quad -5.0 + (-3.1) = -8.1
\]
- Answer: \(-3.1, -5.0, -8.1\).

13. (m) \(\frac{1}{11}, \frac{3}{11}, \frac{4}{11}, \frac{7}{11}, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
\frac{4}{11} = \frac{3}{11} + \frac{1}{11}, \quad \frac{7}{11} = \frac{4}{11} + \frac{3}{11}
\]
- Next terms:
\[
\frac{7}{11} + \frac{4}{11} = \frac{11}{11} = 1, \quad 1 + \frac{7}{11} = \frac{11}{11} + \frac{7}{11} = \frac{18}{11}, \quad \frac{18}{11} + 1 = \frac{18}{11} + \frac{11}{11} = \frac{29}{11}
\]
- Answer: \(1, \frac{18}{11}, \frac{29}{11}\).

14. (n) \(\frac{5}{6}, \frac{11}{12}, \frac{7}{4}, \frac{8}{3}, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
\frac{7}{4} = \frac{11}{12} + \frac{5}{6}, \quad \frac{8}{3} = \frac{7}{4} + \frac{11}{12}
\]
- Convert all fractions to a common denominator (12):
\[
\frac{5}{6} = \frac{10}{12}, \quad \frac{11}{12} = \frac{11}{12}, \quad \frac{7}{4} = \frac{21}{12}, \quad \frac{8}{3} = \frac{32}{12}
\]
- Next terms:
\[
\frac{32}{12} + \frac{21}{12} = \frac{53}{12}, \quad \frac{53}{12} + \frac{32}{12} = \frac{85}{12}, \quad \frac{85}{12} + \frac{53}{12} = \frac{138}{12} = \frac{23}{2}
\]
- Answer: \(\frac{53}{12}, \frac{85}{12}, \frac{23}{2}\).

15. (o) \(-\frac{1}{5}, \frac{1}{2}, \frac{3}{10}, \frac{4}{5}, \ldots\)
- Pattern: Each term is the sum of the two preceding terms.
\[
\frac{3}{10} = \frac{1}{2} + \left(-\frac{1}{5}\right), \quad \frac{4}{5} = \frac{3}{10} + \frac{1}{2}
\]
- Convert all fractions to a common denominator (10):
\[
-\frac{1}{5} = -\frac{2}{10}, \quad \frac{1}{2} = \frac{5}{10}, \quad \frac{3}{10} = \frac{3}{10}, \quad \frac{4}{5} = \frac{8}{10}
\]
- Next terms:
\[
\frac{8}{10} + \frac{3}{10} = \frac{11}{10}, \quad \frac{11}{10} + \frac{8}{10} = \frac{19}{10}, \quad \frac{19}{10} + \frac{11}{10} = \frac{30}{10} = 3
\]
- Answer: \(\frac{11}{10}, \frac{19}{10}, 3\).

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{(a)} & 5 \\
\text{(b)} & 8 \\
\text{(c)} & F_n = F_{n-1} + F_{n-2} \\
\text{(d)} & 16, 26, 42 \\
\text{(e)} & 24, 39, 63 \\
\text{(f)} & 32, 52, 84 \\
\text{(g)} & 100, 162, 262 \\
\text{(h)} & 46, 75, 121 \\
\text{(i)} & 9, 16, 25 \\
\text{(j)} & 250, 405, 655 \\
\text{(k)} & -11, -18, -29 \\
\text{(l)} & 10.5, 17.1, 27.6 \\
\text{(m)} & 7.18, 11.93, 19.11 \\
\text{(n)} & -6.9, -9.8, -16.7 \\
\text{(o)} & -3.1, -5.0, -8.1 \\
\text{(p)} & 1, \frac{18}{11}, \frac{29}{11} \\
\text{(q)} & \frac{53}{12}, \frac{85}{12}, \frac{23}{2} \\
\text{(r)} & \frac{11}{10}, \frac{19}{10}, 3 \\
\end{array}
}
\]