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Chapter 11: Problem 26

Solve. Find the sum of the first five terms of the sequence \(-2,-6\) \(-18, \ldots\)

Short Answer

Expert verified
The sum of the first five terms is -242.

Step by step solution

01

Identify the Type of Sequence

The given sequence is \(-2, -6, -18, \ldots\). To determine the type of sequence, check the ratio between terms. \(-6 \div (-2) = 3\) and \(-18 \div (-6) = 3\). Since both ratios are equal, this is a geometric sequence with a common ratio \(r = 3\).
02

Find the first term and general formula

The first term of the sequence is \(a_1 = -2\). The nth term formula for a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\). For this sequence, \(a_n = -2 \cdot 3^{(n-1)}\).
03

Use the formula for the sum of first n terms

The sum of the first \(n\) terms of a geometric sequence is given by \(S_n = a_1 \cdot \frac{r^n - 1}{r - 1}\). We substitute \(a_1 = -2\), \(r = 3\), and \(n = 5\).
04

Calculate the sum of the first five terms

Substitute the values into the sum formula: \(S_5 = -2 \cdot \frac{3^5 - 1}{3 - 1}\). Calculate \(3^5 = 243\), so \(S_5 = -2 \cdot \frac{243 - 1}{2}\). Simplify: \(-2 \cdot \frac{242}{2} = -2 \cdot 121 = -242\).

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Key Concepts

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

Sequence Terms
In a geometric sequence, every term is derived by multiplying the previous term by a specific number called the common ratio. Each term follows a pattern that makes it predictable and easy to find other terms once a few are known. For example, in the sequence
  • \(-2, -6, -18, \ldots\)
the first term is \(-2\). The second term is formed by multiplying the first term by the common ratio, which in this case, is 3. Therefore, the second term is \(-6\) because \(-2 \times 3 = -6\). Similarly, the third term \(-18\) is obtained from the second term by multiplying \(-6\) by the same common ratio 3, giving \(-6 \times 3 = -18\). With this approach, any term in the sequence can be found.
Common Ratio
The common ratio is a pivotal characteristic of a geometric sequence. It is the number you consistently multiply by to get from one term to the next. In our sequence example, the initial terms \(-2, -6,\) and \(-18\) are used to determine this ratio. Calculate it by dividing each term by its preceding term:
  • \(-6 \div (-2) = 3\)
  • \(-18 \div (-6) = 3\)
If these calculations are consistent across other terms, the common ratio remains the same. In this situation, the common ratio is 3, reflecting the uniformity that is key in geometric progressions.
Series Sum
In geometric sequences, you might need to calculate the sum of a certain number of terms. Fortunately, a formula assists in finding this value without computing each term individually. The sum of the first \(n\) terms, denoted as \(S_n\), is given by the formula:\[S_n = a_1 \cdot \frac{r^n - 1}{r - 1} \]where:
  • \(a_1\) is the first term
  • \(r\) is the common ratio
  • \(n\) is the number of terms
In our example, substituting \(a_1 = -2\), \(r = 3\), and \(n = 5\) into the formula, you perform the calculation\[S_5 = -2 \cdot \frac{3^5 - 1}{3 - 1}\]Evaluating \(3^5\) equals 243, so the sum becomes\[-2 \cdot \frac{243 - 1}{2} = -2 \cdot 121 = -242\]This illustrates how effective the series sum formula is in geometric sequences.
Nth Term Formula
The nth term of a geometric sequence can be directly calculated using a specific formula. This formula allows you to find any term in the series without having to go through each term before it.The formula is expressed as:\[a_n = a_1 \cdot r^{(n-1)}\]Here:
  • \(a_1\) is the first term of the sequence
  • \(r\) is the common ratio
  • \(n\) signifies the position of the term in the sequence
Applying this to our example with \(a_1 = -2\) and \(r = 3\), finding the 5th term (\(n=5\)) would work out as follows:\[a_5 = -2 \cdot 3^{(5-1)} = -2 \cdot 81 = -162\]This efficient formula provides a practical way to calculate any term's value within a geometric sequence, highlighting its consistency and cohesiveness.

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Most popular questions from this chapter

Evaluate. See Section 1.3 $$ \frac{8-1}{8+1}+\frac{8-2}{8+2}+\frac{8-3}{8+3} $$

Use the partial sum formula to find the partial sum of the given arithmetic or geometric sequence. See Examples 1 and 4 Find the sum of the first seven terms of the arithmetic sequence \(-7,-11,-15, \ldots\)

Determine whether the sequence is arithmetic, geometric,or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. \(a_{n}=n-3\)

Solve. See Example 2. Find the sum of the first five negative odd integers.

A bacterial colony begins with 100 bacteria and grows according to the sequence defined by \(a_{n}=100 \cdot 2^{n-1}\) where \(n\) is the number of 6 -hour periods. Find the total number of bacteria there will be after 24 hours.
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