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Chapter 11: Problem 23

Find \(S_{n}\) for each geometric series described. $$ a_{1}=5, r=3, n=12 $$

Short Answer

Expert verified
The sum of the series, \( S_{12} \), is 1,328,600.

Step by step solution

01

Identify the formula for the sum of a geometric series

A geometric series is the sum of the terms of a geometric sequence. The formula to find the sum of the first \( n \) terms is given by: \[ S_n = a_1 \frac{r^n - 1}{r - 1} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
02

Substitute the given values into the formula

Now, substitute the given values \( a_1 = 5 \), \( r = 3 \), and \( n = 12 \) into the formula: \[ S_{12} = 5 \frac{3^{12} - 1}{3 - 1} \]
03

Calculate \( 3^{12} \)

Compute \( 3^{12} \), which is a part of the formula:\[ 3^{12} = 531441 \]
04

Substitute back into the formula

Substitute \( 3^{12} = 531441 \) back into the formula:\[ S_{12} = 5 \frac{531441 - 1}{2} \]
05

Simplify the expression inside the fraction

Subtract 1 from 531441:\[ 531441 - 1 = 531440 \]
06

Simplify the fraction

Perform the division inside the fraction:\[ \frac{531440}{2} = 265720 \]
07

Multiply by the first term

Finally, multiply the resulting fraction by the first term:\[ S_{12} = 5 \times 265720 = 1328600 \]

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Key Concepts

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

Sum of Geometric Series
The sum of a geometric series refers to the total when you add up all the terms in a geometric sequence. For understanding this plainly, a geometric series is built on terms that follow a specific pattern. In our scenario, the first term of the series is 5, and each succeeding term is obtained by multiplying the previous term by a constant value known as the "common ratio". To figure out the sum, we use the formula:\[S_n = a_1 \frac{r^n - 1}{r - 1}\]Where:
  • \(S_n\) is the sum of the series.
  • \(a_1\) is the initial term, here it's 5.
  • \(r\) stands for the common ratio, which is 3.
  • \(n\) represents the number of terms, which in this case is 12.
This formula helps us aggregate the values resulting from the repetitive multiplication described in the sequence.
Common Ratio
The common ratio is a key element in defining a geometric sequence. It is the factor by which we multiply each term in the sequence to get to the next term. In the presented exercise, the common ratio \( r \) is 3. To observe this in action:
  • Start with the first term, 5.
  • Multiply 5 by 3 to get 15.
  • Multiply 15 by 3 to get 45.
  • This pattern continues repeatedly for all terms.
Because the ratio remains consistent across the sequence, it ensures the series maintains its geometric nature. Consequently, understanding the common ratio is essential for solving problems involving geometric series.
Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by the common ratio. In other words, there's a fixed number you multiply by to get from one term to the next. For our example in the exercise, the sequence begins at 5 and continues with terms created by multiplying by 3:
  • 5, 15, 45, 135, ...
As observed, by continually multiplying by the common ratio, 3, the sequence develops incrementally. A geometric sequence like this one is exponential in nature, showing rapid growth when compared to arithmetic sequences where we add a fixed number instead of multiplying.
Terms of the Series
In the context of a geometric series, the "terms" refer to the numbers in the sequence that we are summing up. The first term is denoted as \( a_1 \), followed by terms formed through multiplying by the common ratio. For the exercise in discussion, the terms are identified and collaborated.For instance, given:
  • The first term \( a_1 = 5 \).
  • The second term is \( a_1 \times 3 = 15 \).
  • The third term extends to \( 15 \times 3 = 45 \).
This continued multiplication outlines all the terms within your desired range. Here, we compute up to the 12th term. These terms are independently significant as they build up the sequence and collectively contribute to the sum of the series.

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