Let's carefully analyze and solve each part of the question step by step.
### Problem 1
Rule for the sequence: Each number is the sum of the previous two numbers. We start with the numbers 0 and 1.
Solution: Let's build the sequence of 10 numbers:
1. Start with 0 and 1.
2. The next number is the sum of the previous two numbers:
- 0 + 1 = 1
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 3 + 5 = 8
- 5 + 8 = 13
- 8 + 13 = 21
- 13 + 21 = 34
So, the sequence becomes:
[tex]\[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 \][/tex]
### Problem 2
A sequence starts as [tex]\(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \cdots\)[/tex]
a. Rule: This sequence follows a geometric progression where every term is half of the preceding term. Alternatively, we could say each term [tex]\( T_n \)[/tex] can be expressed as [tex]\( T_n = \frac{1}{2^n} \)[/tex].
b. Next 3 terms:
- [tex]\( T_4 = \frac{1}{2^4} = \frac{1}{16} \)[/tex]
- [tex]\( T_5 = \frac{1}{2^5} = \frac{1}{32} \)[/tex]
- [tex]\( T_6 = \frac{1}{2^6} = \frac{1}{64} \)[/tex]
Thus, the next three terms are:
[tex]\[ \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \][/tex]
### Problem 3
A sequence follows the rule: multiply the previous number by -2 and add 3. The fourth term in the sequence is -7.
a. Finding the next 3 terms:
Starting from the fourth term:
- 4th term: [tex]\(-7\)[/tex]
Calculate the next terms using the rule:
- 5th term: [tex]\((-7 \times -2) + 3 = 14 + 3 = 17\)[/tex]
- 6th term: [tex]\((17 \times -2) + 3 = -34 + 3 = -31\)[/tex]
- 7th term: [tex]\((-31 \times -2) + 3 = 62 + 3 = 65\)[/tex]
So, the next three terms are:
[tex]\[ 17, -31, 65 \][/tex]
b. Finding the 3 terms that came before [tex]\(-7\)[/tex]:
Starting from the fourth term:
- 4th term: [tex]\(-7\)[/tex]
Calculate the previous terms in reverse using the rule:
Reverse calculation:
- To find the 3rd term, let [tex]\( x \)[/tex] be the value, solving for [tex]\( x \)[/tex]:
[tex]\((-7 - 3) / -2 = x \Rightarrow -10 / -2 = x \Rightarrow x = 5\)[/tex]
- To find the 2nd term, let [tex]\( y \)[/tex] be the value, solving for [tex]\( y \)[/tex]:
[tex]\((5 - 3) / -2 = y \Rightarrow 2 / -2 = y \Rightarrow y = -1\)[/tex]
- To find the 1st term, let [tex]\( z \)[/tex] be the value, solving for [tex]\( z \)[/tex]:
[tex]\((-1 - 3) / -2 = z \Rightarrow -4 / -2 = z \Rightarrow z = 2\)[/tex]
So, the previous three terms are:
[tex]\[ 2, -1, 5 \][/tex]
Summary of sequences derived:
- The sequence with 10 numbers: [tex]\([0, 1, 1, 2, 3, 5, 8, 13, 21, 34]\)[/tex]
- The next three terms for the geometric sequence: [tex]\(\left( \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \right)\)[/tex]
- The next three terms for the sequence given with fourth term -7: [tex]\([17, -31, 65]\)[/tex]
- The three terms before -7: [tex]\([2, -1, 5]\)[/tex]
