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Chapter 11: Problem 90

Write a recursion formula for each sequence. $$30,25,20,15, \dots$$

Short Answer

Expert verified
The recursion formula is \(a_{n+1} = a_n - 5\) with \(a_1 = 30\).

Step by step solution

01

Identify the Pattern

Examine the sequence to identify the pattern or rule that governs the progression. Notice that each term is decreasing by 5 from the previous term.
02

Represent the Sequence

Let the sequence be denoted by \({a_n}\), where \({a_1 = 30, a_2 = 25, a_3 = 20, a_4 = 15, \dots}\). Start by assigning the first term: \(a_1 = 30\).
03

Write the General Recursion Formula

The pattern suggests each term is obtained by subtracting 5 from the previous term. Therefore, the recursion formula is: \(a_{n+1} = a_n - 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

arithmetic sequences
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
An example of an arithmetic sequence is given in the exercise: 30, 25, 20, 15, \dots.
Each term decreases by 5, so the common difference (d) is -5.
In general terms, if \(a_n\) is the nth term, the initial term is \(a_1\) and common difference is d, the sequence can be defined recursively as:
\(a_{n+1} = a_n + d\).
Given d is -5, the recursive formula for the sequence becomes:
\(a_{n+1} = a_n - 5\).
precalculus patterns
Understanding patterns is a crucial part of precalculus. A pattern represents a set of numbers or objects that follow a specific rule. Recognizing these rules allows us to predict future terms.
In the given exercise, we notice the consistent subtraction of 5 from each term to obtain the next. This recognition leads to forming a precise recursion formula.
A recursion formula helps express each term based on the previous one, making calculations straightforward. For mathematical sequences, especially in precalculus, this formula is vital for predicting large sequences efficiently.
mathematical progression
Mathematical progression refers to the systematic and predictable growth or reduction in a sequence of numbers. Types of progressions include arithmetic, geometric, and harmonic.
The problem demonstrates an arithmetic progression (AP), where the change between terms is constant. This constant change simplifies to a recursive relationship. For any progression, identifying the type (arithmetic, geometric) is crucial for setting up correct formulas.
Through the exercise, we see how mathematical progression aids in structuring a sequence formula:\(a_{n+1} = a_n - 5\). The capability to build such formulas enhances mathematical understanding and problem-solving skills.

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