Math Problem Statement
90,30,10,10/3 Write a recursive formula for each sequence.
, 8, 13, 18, 23…
, 2, −8, 32, −128…
=a18
=a12
=an5for≥n2
=an4for≥n2
Solution
Sequence 1: 90, 30, 10, 10/3, ...
Let's denote the -th term of the sequence by .
First term:
General pattern: Each term is one-third of the previous term.
Recursive formula:
a_1 = 90 \\ a_{n+1} = \frac{a_n}{3} \quad \text{for } n \geq 1 \end{cases}$$ ### Sequence 2: 8, 13, 18, 23, ... Let's denote the $$n$$-th term of the sequence by $$b_n$$. **First term:** $$b_1 = 8$$ **General pattern:** Each term increases by 5 from the previous term. $$b_{n+1} = b_n + 5$$ **Recursive formula:** $$\begin{cases} b_1 = 8 \\ b_{n+1} = b_n + 5 \quad \text{for } n \geq 1 \end{cases}$$ ### Sequence 3: 2, −8, 32, −128, ... Let's denote the $$n$$-th term of the sequence by $$c_n$$. **First term:** $$c_1 = 2$$ **General pattern:** Each term is multiplied by -4 from the previous term. $$c_{n+1} = -4 \cdot c_n$$ **Recursive formula:** $$\begin{cases} c_1 = 2 \\ c_{n+1} = -4 \cdot c_n \quad \text{for } n \geq 1 \end{cases}$$ ### Sequence 4: $$*a*18$$ Since the sequence is given as "*a*18," let's assume this is a constant sequence where each term is 18. Let's denote the $$n$$-th term of the sequence by $$d_n$$. **First term:** $$d_1 = 18$$ **General pattern:** Each term is 18. $$d_{n+1} = d_n$$ **Recursive formula:** $$\begin{cases} d_1 = 18 \\ d_{n+1} = d_n \quad \text{for } n \geq 1 \end{cases}$$ ### Sequence 5: $$*a*12$$ Similarly, let's assume this is a constant sequence where each term is 12. Let's denote the $$n$$-th term of the sequence by $$e_n$$. **First term:** $$e_1 = 12$$ **General pattern:** Each term is 12. $$e_{n+1} = e_n$$ **Recursive formula:** $$\begin{cases} e_1 = 12 \\ e_{n+1} = e_n \quad \text{for } n \geq 1 \end{cases}$$ ### Sequence 6: $$*an*5$$ for $$n \geq 2$$ This appears to be an arithmetic sequence with a common difference of 5, starting from an unspecified initial term $$a$$. Let's denote the $$n$$-th term of the sequence by $$f_n$$. **First term:** $$f_1 = a$$ **General pattern:** Each term increases by 5 from the previous term. $$f_{n+1} = f_n + 5$$ **Recursive formula:** $$\begin{cases} f_1 = a \\ f_{n+1} = f_n + 5 \quad \text{for } n \geq 1 \end{cases}$$ ### Sequence 7: $$*an*4$$ for $$n \geq 2$$ This appears to be a geometric sequence with a common ratio of 4, starting from an unspecified initial term $$a$$. Let's denote the $$n$$-th term of the sequence by $$g_n$$. **First term:** $$g_1 = a$$ **General pattern:** Each term is multiplied by 4 from the previous term. $$g_{n+1} = 4 \cdot g_n$$ **Recursive formula:** $$\begin{cases} g_1 = a \\ g_{n+1} = 4 \cdot g_n \quad \text{for } n \geq 1 \end{cases}$$ --- Do you want any details or have any questions? Here are some related questions: 1. How do you determine if a sequence is arithmetic or geometric? 2. What is the difference between a common difference and a common ratio? 3. Can a sequence be both arithmetic and geometric? 4. How do you find the explicit formula for a geometric sequence? 5. What is the significance of the first term in a recursive sequence? 6. How can you convert a recursive formula to an explicit formula? 7. What are some real-life applications of recursive sequences? 8. How do you identify patterns in non-arithmetic and non-geometric sequences? **Tip:** Always check if the terms of a sequence follow an arithmetic or geometric pattern before writing the recursive formula. This will help you identify the correct rule for the sequence.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Recursive Sequences
Arithmetic Sequences
Geometric Sequences
Formulas
Recursive formula for arithmetic sequences: a_{n+1} = a_n + d
Recursive formula for geometric sequences: a_{n+1} = r \cdot a_n
Theorems
-
Suitable Grade Level
High School
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