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Sagot :
Sure! Let's tackle each part of the problem step by step.
### Part 1: Fibonacci Sequence
The first part of the problem asks us to generate a sequence where each number is the sum of the previous two numbers, starting from 0 and 1. This is a well-known sequence called the Fibonacci sequence. We need to generate the first 10 numbers in this sequence.
#### Steps to Generate the Fibonacci Sequence:
1. Start with the initial two numbers: 0 and 1.
2. To find the next number in the sequence, sum the last two numbers in the sequence.
3. Repeat this process until we have 10 numbers in the sequence.
Let's display the Fibonacci sequence by iteratively summing the last two numbers:
- Step 1: Start with [0, 1]
- Step 2: 0 + 1 = 1, so the sequence becomes [0, 1, 1]
- Step 3: 1 + 1 = 2, so the sequence becomes [0, 1, 1, 2]
- Step 4: 1 + 2 = 3, so the sequence becomes [0, 1, 1, 2, 3]
- Step 5: 2 + 3 = 5, so the sequence becomes [0, 1, 1, 2, 3, 5]
- Step 6: 3 + 5 = 8, so the sequence becomes [0, 1, 1, 2, 3, 5, 8]
- Step 7: 5 + 8 = 13, so the sequence becomes [0, 1, 1, 2, 3, 5, 8, 13]
- Step 8: 8 + 13 = 21, so the sequence becomes [0, 1, 1, 2, 3, 5, 8, 13, 21]
- Step 9: 13 + 21 = 34, so the sequence becomes [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
So the first 10 numbers in the Fibonacci sequence are:
[tex]\[ [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] \][/tex]
### Part 2: Geometric Sequence
The second part of the problem involves a sequence that starts with [tex]\( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \)[/tex].
#### Part 2a: Identify the Rule
We need to identify a rule that the sequence follows. Observing the given sequence:
[tex]\[ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \][/tex]
We can see that each term is obtained by dividing the previous term by 2. In other words, each term is half of the previous term.
#### Part 2b: Find the Next 3 Terms
Using the identified rule (each term is half of the previous term), let's find the next three terms in the sequence:
1. Starting term: [tex]\( \frac{1}{8} \)[/tex]
2. The next term: [tex]\( \frac{1}{8} \div 2 = \frac{1}{16} \)[/tex]
3. The term after that: [tex]\( \frac{1}{16} \div 2 = \frac{1}{32} \)[/tex]
4. The term following that: [tex]\( \frac{1}{32} \div 2 = \frac{1}{64} \)[/tex]
So, the next three terms in the sequence are:
[tex]\[ \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \][/tex]
Therefore, the result of our work is as follows:
1. The first 10 numbers of the Fibonacci sequence are:
[tex]\[ [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] \][/tex]
2. The next three terms in the given sequence are:
[tex]\[ \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \][/tex]
### Part 1: Fibonacci Sequence
The first part of the problem asks us to generate a sequence where each number is the sum of the previous two numbers, starting from 0 and 1. This is a well-known sequence called the Fibonacci sequence. We need to generate the first 10 numbers in this sequence.
#### Steps to Generate the Fibonacci Sequence:
1. Start with the initial two numbers: 0 and 1.
2. To find the next number in the sequence, sum the last two numbers in the sequence.
3. Repeat this process until we have 10 numbers in the sequence.
Let's display the Fibonacci sequence by iteratively summing the last two numbers:
- Step 1: Start with [0, 1]
- Step 2: 0 + 1 = 1, so the sequence becomes [0, 1, 1]
- Step 3: 1 + 1 = 2, so the sequence becomes [0, 1, 1, 2]
- Step 4: 1 + 2 = 3, so the sequence becomes [0, 1, 1, 2, 3]
- Step 5: 2 + 3 = 5, so the sequence becomes [0, 1, 1, 2, 3, 5]
- Step 6: 3 + 5 = 8, so the sequence becomes [0, 1, 1, 2, 3, 5, 8]
- Step 7: 5 + 8 = 13, so the sequence becomes [0, 1, 1, 2, 3, 5, 8, 13]
- Step 8: 8 + 13 = 21, so the sequence becomes [0, 1, 1, 2, 3, 5, 8, 13, 21]
- Step 9: 13 + 21 = 34, so the sequence becomes [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
So the first 10 numbers in the Fibonacci sequence are:
[tex]\[ [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] \][/tex]
### Part 2: Geometric Sequence
The second part of the problem involves a sequence that starts with [tex]\( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \)[/tex].
#### Part 2a: Identify the Rule
We need to identify a rule that the sequence follows. Observing the given sequence:
[tex]\[ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \][/tex]
We can see that each term is obtained by dividing the previous term by 2. In other words, each term is half of the previous term.
#### Part 2b: Find the Next 3 Terms
Using the identified rule (each term is half of the previous term), let's find the next three terms in the sequence:
1. Starting term: [tex]\( \frac{1}{8} \)[/tex]
2. The next term: [tex]\( \frac{1}{8} \div 2 = \frac{1}{16} \)[/tex]
3. The term after that: [tex]\( \frac{1}{16} \div 2 = \frac{1}{32} \)[/tex]
4. The term following that: [tex]\( \frac{1}{32} \div 2 = \frac{1}{64} \)[/tex]
So, the next three terms in the sequence are:
[tex]\[ \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \][/tex]
Therefore, the result of our work is as follows:
1. The first 10 numbers of the Fibonacci sequence are:
[tex]\[ [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] \][/tex]
2. The next three terms in the given sequence are:
[tex]\[ \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \][/tex]
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