Math Problem Statement

Recursive formula for the sequence represented by the explicit formula a n=1/5(7)n-1

Solution

To find the recursive formula for the sequence represented by the explicit formula:

an=157n1,a_n = \frac{1}{5} \cdot 7^{n-1},

we observe the relationship between consecutive terms.

Step-by-step process:

  1. Explicit formula: an=157n1.a_n = \frac{1}{5} \cdot 7^{n-1}.

  2. Compute the ratio of consecutive terms: an+1an=157n157n1=7.\frac{a_{n+1}}{a_n} = \frac{\frac{1}{5} \cdot 7^n}{\frac{1}{5} \cdot 7^{n-1}} = 7. This shows that each term is 7 times the previous term.

  3. Recursive formula: Using the ratio between consecutive terms, we can write the recursive formula as: an=7an1,for n2,a_n = 7 \cdot a_{n-1}, \quad \text{for } n \geq 2, with the initial condition: a1=15.a_1 = \frac{1}{5}.

Final Recursive Formula:

an=7an1,where a1=15.a_n = 7 \cdot a_{n-1}, \quad \text{where } a_1 = \frac{1}{5}.

Would you like a detailed explanation of how recursive formulas are derived or their applications?


Related Questions:

  1. How do explicit and recursive formulas differ in practice?
  2. What is the general method to convert between explicit and recursive formulas?
  3. How can we determine the nn-th term directly using a recursive formula?
  4. Can a sequence have more than one recursive formula representation?
  5. What happens if we change the base of the exponential term in the sequence?

Tip:

To verify a recursive formula, test it with small values of nn to ensure it correctly generates the sequence from the explicit formula.

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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Recursive Formulas
Exponential Functions

Formulas

Explicit formula: a_n = (1/5)(7)^(n-1)
Recursive formula: a_n = 7 * a_(n-1)

Theorems

Properties of Exponential Functions
Recursive Sequence Relations

Suitable Grade Level

Grades 9-12