Let's analyze the given sequences to determine if they are indeed the same.
We begin with the examined sequence: [tex]\(a_1=2\)[/tex], [tex]\(a_2=4\)[/tex], [tex]\(a_3=8\)[/tex], [tex]\(a_4=16\)[/tex], [tex]\(a_5=32\)[/tex], ...
Now, let's consider the recursive definition for this sequence:
- [tex]\(a_1=2\)[/tex]
- For [tex]\(n>1\)[/tex], [tex]\(a_n = 2 \cdot a_{n-1}\)[/tex]
We'll calculate the terms of the sequence using this recursive definition step by step.
1. [tex]\(a_1 = 2\)[/tex]
2. [tex]\(a_2 = 2 \cdot a_1 = 2 \cdot 2 = 4\)[/tex]
3. [tex]\(a_3 = 2 \cdot a_2 = 2 \cdot 4 = 8\)[/tex]
4. [tex]\(a_4 = 2 \cdot a_3 = 2 \cdot 8 = 16\)[/tex]
5. [tex]\(a_5 = 2 \cdot a_4 = 2 \cdot 16 = 32\)[/tex]
Now, compare the terms from the recursive definition with the given sequence:
- Given sequence: [tex]\(2, 4, 8, 16, 32\)[/tex]
- Calculated sequence: [tex]\(2, 4, 8, 16, 32\)[/tex]
Since the terms of the sequences match exactly, we can conclude that the given sequence is, indeed, the same as the sequence defined recursively.
Hence, the answer is A. True.
