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Chapter 11: Problem 169

Write a recursive formula for each geometric sequence. \(a_{n}=\\{-1,5,-25,125, \ldots\\}\)

Short Answer

Expert verified
The recursive formula is \(a_1 = -1\); \(a_n = a_{n-1} \times (-5)\) for \(n > 1\).

Step by step solution

01

Identify the First Term

The first term of the sequence, denoted as \(a_1\), is \(-1\).
02

Determine the Common Ratio

To find the common ratio \(r\), divide the second term by the first: \(r = \frac{5}{-1} = -5\). Verify this by checking that each term is the product of the previous term and \(-5\).
03

Define the Recursive Formula

A recursive formula for a geometric sequence provides any term based on the previous term. Since we know the first term \(a_1 = -1\) and the common ratio \(r = -5\), the formula is \(a_n = a_{n-1} \times (-5)\) for \(n > 1\).

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Key Concepts

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

Geometric Sequences
In mathematics, a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number, called the common ratio. For example, in the sequence \(-1, 5, -25, 125, \ldots\), each term is created by multiplying the previous term by \(-5\). Geometric sequences are fundamental concepts that often arise in various mathematical and real-world applications.
They are characterized by a constant ratio between successive terms, which makes them predictable and easy to model.
  • Every geometric sequence has a clear pattern, driven by multiplication.
  • The terms grow (or shrink) exponentially, making such sequences intriguing for those studying them.
  • Understanding geometric sequences can help in solving diverse problems, from computer science to finance.
Overall, recognizing and working with geometric sequences can equip you with skills to tackle complex sequences and patterns with ease.
Common Ratio
The common ratio, often symbolized as \(r\), is a pivotal component of any geometric sequence. It is calculated by dividing any term in the sequence by its preceding term. For instance, in our sequence \(-1, 5, -25, 125, \ldots\), the common ratio \(r\) can be determined by calculating \(\frac{5}{-1} = -5\). This ratio reveals the fixed factor by which each term multiplies to produce the next term.
Knowing the common ratio is essential because:
  • It helps in determining the recursive formula of the sequence.
  • It defines the nature of the sequence, whether it grows or shrinks.
  • Being negative or positive, it affects the sequence's oscillation or steady increase.
As you explore further, you'll find that understanding the common ratio significantly aids in predicting sequence patterns and extends into solving various mathematical challenges.
First Term
The first term of a geometric sequence, denoted as \(a_1\), is the starting point of the sequence from which all other terms unfold. In our example, \(a_1 = -1\). Identifying the first term is crucial in both defining the sequence and establishing the recursive formula.
Here's why the first term is important:
  • It sets the initial value from which subsequent terms are derived using the common ratio.
  • Understanding it helps establish the context or baseline of a sequence.
  • In recursion, it serves as the 'seed' needed to grow the entire sequence.
In summary, the first term is like the anchor of the entire geometric sequence. Without it, constructing or applying recursive formulas would be impossible, as there would be no initial reference point.

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