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Chapter 8: Problem 27

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 11 terms of the geometric sequence: $$3,-6,12,-24, \dots$$

Short Answer

Expert verified
After simplifying the expression in Step 3, one finds that the sum of the first 11 terms of the given geometric sequence is -12285.

Step by step solution

01

Identify the first term and the common ratio

In the given sequence, the first term \(a\) is 3. We obtain the common ratio by dividing the second term by the first term. In this case, the common ratio \(r\) is -6/3 = -2.
02

Apply the formula for the sum of the first n terms of a geometric sequence

The formula to calculate the sum of a geometric series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). Substituting the values we get \(S_{11} = \frac{3 (1 - (-2)^{11})}{1 - (-2)}\).
03

Simplify the expression

Arithmetic simplification of this expression will give you the sum of the first 11 terms.

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Key Concepts

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

Understanding the Common Ratio
In a geometric sequence, the common ratio is a fundamental concept. It tells us how to move from one term to another. To find the common ratio, simply divide a term by the previous term.
For the sequence given: \(3, -6, 12, -24, \dots\), divide the second term by the first term:
  • \(-6 \div 3 = -2\)
Thus, the common ratio \(r\) is \(-2\).
This ratio multiplies each term to get the next, making the sequence predictable and easy to analyze.
Identifying the First Term
The first term in a sequence sets the stage for everything that follows. It is denoted as \(a\) in mathematical formulas. Here, the initial value is simply the very first number in the sequence:
  • For the given sequence: \(3, -6, 12, -24, \dots\), the first term \(a\) is \(3\).
This starting point is crucial, as it helps determine the entire course of the sequence when combined with the common ratio.
Calculating the Sum of a Geometric Series
To find the sum of the first \(n\) terms of a geometric sequence, use a specific formula:
\[S_n = \frac{a(1 - r^n)}{1 - r}\]
This equation considers both the first term \(a\) and the common ratio \(r\).
For our sequence, we want the sum of the first 11 terms, so:
  • \(a = 3\)
  • \(r = -2\)
Plug these into the formula:
\[S_{11} = \frac{3(1 - (-2)^{11})}{1 - (-2)}\]
This setup allows you to compute the sum quickly and efficiently.
Arithmetic Simplification
Arithmetic simplification is the crucial final step to solve the problem. Simplify the expression from the sum formula to find the exact sum over the desired number of terms.
Start with:
\[S_{11} = \frac{3(1 - (-2)^{11})}{1 - (-2)}\]
Begin by simplifying row by row:
  • Calculate \((-2)^{11}\): This gives a large negative number.
  • Insert this value back into the equation and simplify the numerator and the denominator separately.
The final simplification yields the sum of the first 11 terms. Careful arithmetic ensures a correct and precise outcome.

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