Math Problem Statement
Find the recursive definition of the sequence: 11, 22, 44, 88.
Solution
The given sequence is . To find the recursive definition, we observe how each term relates to the previous term. Here's the analysis:
Step-by-step Analysis:
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Finding the pattern:
The second term () is twice the first term (), the third term () is twice the second term, and the fourth term () is twice the third term.
Mathematically, . -
Initial condition:
The sequence starts with .
Recursive Formula:
a_n = 2a_{n-1} & \text{for } n \geq 2, \\ a_1 = 11. \end{cases}$$ ### Comparing with the choices in the image: - **First option ($$a_n = a_{n-1} + 2$$)**: Incorrect because the sequence is not increasing by a constant difference but by multiplication. - **Second option ($$a_n = 2a_{n-1}$$)**: Correct as explained above. - **Third option ($$a_n = 11a_{n-1}$$)**: Incorrect because multiplying by $$11$$ does not produce the correct sequence. Thus, the correct recursive definition is: $$\begin{cases} a_n = 2a_{n-1}, & a_1 = 11. \end{cases}$$ --- Would you like a deeper explanation or further questions to explore this topic? Here are some related questions to consider: 1. How do you identify a recursive formula for a given sequence? 2. What is the explicit (non-recursive) formula for this sequence? 3. How do arithmetic sequences differ from geometric sequences? 4. How can you determine if a sequence is linear or exponential? 5. Can you extend this sequence to find the 6th or 7th term? **Tip:** To find the $$n$$-th term of a geometric sequence directly, use the formula $$a_n = a_1 \cdot r^{n-1}$$, where $$r$$ is the common ratio.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Sequences and Series
Recursion
Geometric Progressions
Formulas
Recursive formula: a_n = r * a_(n-1)
Common ratio for geometric sequence
Theorems
Geometric Sequence Properties
Suitable Grade Level
Grades 8-10