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Write an explicit formula that represents the sequence defined by the following recursive formula: a1 -7 and an = An-1 = - 10

Write an explicit formula that represents the sequence defined by the following recursive formula: a1 -7 and an = An-1 = - 10

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter14: Sequences And Mathematical Induction
Section14.1: Arithmetic Sequences
Problem 79PS
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### Recursive Sequences in Mathematics In mathematics, a sequence is an ordered list of numbers following a particular pattern. Recursive sequences are defined using recurrence relations, which express each term as a function of the preceding terms. Our goal is to derive an explicit formula for the sequence from its recursive definition. #### Problem Statement Here is a problem involving a recursive sequence. The task is to write an explicit formula that represents the sequence defined by the following recursive formula: \[ a_1 = -7 \quad \text{and} \quad a_n = a_{n-1} - 10 \] #### Formulation and Solving 1. **Initial Term**: The first term of the recursive sequence is given by: \[ a_1 = -7 \] 2. **Recursive Formula**: The recursive relationship between the terms is given by: \[ a_n = a_{n-1} - 10 \] This implies that each term is obtained by subtracting 10 from the previous term. #### Finding the Explicit Formula To find a general, non-recursive (explicit) formula for the sequence, we can analyze the pattern: - \[ a_1 = -7 \] - \[ a_2 = a_1 - 10 = -7 - 10 = -17 \] - \[ a_3 = a_2 - 10 = -17 - 10 = -27 \] - \[ a_4 = a_3 - 10 = -27 - 10 = -37 \] From this, we can observe the pattern \(a_n = -7 - 10(n-1)\). By simplifying the expression, we get: \[ a_n = -7 - 10n + 10 = 3 - 10n \] Hence, the explicit formula for the sequence is: \[ a_n = 3 - 10n \] \[ \text{Answer:} \quad a_n = \boxed{\quad \quad \quad } \] *When filling in the blank, enter: \( a_n = 3 - 10n \)* #### Submission Enter the derived formula in the provided answer box and click "Submit Answer" to check for correctness.
Transcribed Image Text:### Recursive Sequences in Mathematics In mathematics, a sequence is an ordered list of numbers following a particular pattern. Recursive sequences are defined using recurrence relations, which express each term as a function of the preceding terms. Our goal is to derive an explicit formula for the sequence from its recursive definition. #### Problem Statement Here is a problem involving a recursive sequence. The task is to write an explicit formula that represents the sequence defined by the following recursive formula: \[ a_1 = -7 \quad \text{and} \quad a_n = a_{n-1} - 10 \] #### Formulation and Solving 1. **Initial Term**: The first term of the recursive sequence is given by: \[ a_1 = -7 \] 2. **Recursive Formula**: The recursive relationship between the terms is given by: \[ a_n = a_{n-1} - 10 \] This implies that each term is obtained by subtracting 10 from the previous term. #### Finding the Explicit Formula To find a general, non-recursive (explicit) formula for the sequence, we can analyze the pattern: - \[ a_1 = -7 \] - \[ a_2 = a_1 - 10 = -7 - 10 = -17 \] - \[ a_3 = a_2 - 10 = -17 - 10 = -27 \] - \[ a_4 = a_3 - 10 = -27 - 10 = -37 \] From this, we can observe the pattern \(a_n = -7 - 10(n-1)\). By simplifying the expression, we get: \[ a_n = -7 - 10n + 10 = 3 - 10n \] Hence, the explicit formula for the sequence is: \[ a_n = 3 - 10n \] \[ \text{Answer:} \quad a_n = \boxed{\quad \quad \quad } \] *When filling in the blank, enter: \( a_n = 3 - 10n \)* #### Submission Enter the derived formula in the provided answer box and click "Submit Answer" to check for correctness.
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