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Chapter 9: Problem 3

The sum of a finite geometric sequence with common ratio \(r \neq 1\) is given by _____.

Short Answer

Expert verified
The sum S of the first n terms of a finite geometric sequence with common ratio \(r \neq 1\) is given by the formula \(S = a \cdot \frac{1 - r^n}{1 - r}\).

Step by step solution

01

Define the relevant geometric sequence parameters

In a geometric sequence, we typically have an initial term (a), a common ratio (r) and the number of terms (n). These parameters are key to finding the sum of the sequence.
02

Apply the formula for the sum of a geometric sequence

The formula to calculate the sum S of the first n terms of a finite geometric sequence is \(S = a \cdot \frac{1 - r^n}{1 - r}\). This is the result that was asked for in the exercise.

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Key Concepts

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

Geometric Sequence
A geometric sequence is an ordered list of numbers. Each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This sequence is distinct for its predictable pattern and multiplication factor.
For example, consider the sequence 3, 6, 12, 24, 48. Here, each term is multiplied by 2 to get the next term. Thus, the common ratio in this case is 2. Geometric sequences can also decrease if the common ratio is a fraction less than 1.
  • The first term is denoted as \(a\).
  • The common ratio is denoted as \(r\).
  • The sequence can be expressed as \(a, ar, ar^2, ar^3, \, ...\).
These properties make geometric sequences easy to identify and calculate, either in full or for specified parts of the total sequence.
Common Ratio
The common ratio in a geometric sequence determines the consistent multiplier between consecutive terms. It's what defines the path of the sequence, whether it's growing, shrinking, or staying constant.
The common ratio \(r\) can be easily determined by dividing any term of the sequence by its previous term. For example, in the sequence 5, 10, 20, 40, \,... the common ratio \(r\) is obtained as \(10 \div 5 = 2\) or \(20 \div 10 = 2\). This verifies that the sequence is indeed geometric with \(r = 2\).
  • The common ratio can be any real number apart from zero.
  • If \(r > 1\), the sequence grows larger with each term.
  • If \(0 < r < 1\), the sequence gets smaller.
  • Negative values of \(r\) will alternate signs among the terms.
Understanding and identifying the common ratio is a crucial step when working with geometric sequences as it affects the entire structure.
Formula for Sum of Geometric Sequence
The formula for the sum of a geometric sequence allows us to find out the total of all terms rapidly without adding each term individually. This is particularly useful for sequences with many terms. The sum of a finite geometric sequence with \(n\) terms can be calculated using:
\[ S = a \cdot \frac{1 - r^n}{1 - r} \]
Where \(\)a\(\) is the first term, \(\)r\(\) is the common ratio, and \(\)n\(\) is the number of terms. This formula is only applicable when \(\)r eq 1\(\).
Let's break down the formula:
  • The numerator \(1 - r^n\) determines the limit reached when the sequence is finite, preventing it from going infinite.
  • The denominator \(1 - r\) adjusts this based on the common ratio, smoothing out scaling effects due to multiplication.
  • Multiplying the fraction by \(a\) scales it, bringing the focus to the actual sequence in context.
By applying this formula, you'll quickly find the summed total of a series of repeating multipliers.

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Most popular questions from this chapter

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