Problem 83 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=1}^{10}8\left(-\dfrac{1}{4}\right)^{i - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is approximately 11.165496826171875

Step 1: Identify the parameters

From the problem, we identify the first term \( a_1 = 8 \), the common ratio \( r = -\dfrac{1}{4} \), and the number of terms \( n = 10 \)

Step 2: Apply the formula

Now we substitute these values into the formula for the sum of a geometric series : \( S_n = a_1 \times \dfrac{1 - r^n}{1 - r} \). We get \( S_{10} = 8 \times \dfrac{1 - (-\dfrac{1}{4})^{10}}{1 - (-\dfrac{1}{4})} \)

Step 3: Calculation

Do the calculations to get \( S_{10} = 8 \times \dfrac{1 - (-0.0000009536743164)}{1 - (-0.25)} = 11.165496826171875 \)

Key Concepts

Geometric Sequence

A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, ... the common ratio is 3, since each term is three times the term before it.

Characteristics of a geometric sequence include:

• The ability to define the nth term using the formula: \( a_n = a_1 \times r^{(n - 1)} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.
• The sequence can increase or decrease depending on whether the common ratio is greater or less than 1, respectively.
• If the common ratio is negative, the sequence will alternate between positive and negative values.

The sequence in our exercise has a first term \( a_1 = 8 \) and a common ratio \( r = -\dfrac{1}{4} \), which indicates that the sequence is decreasing and alternating in sign.

Sum of Geometric Series

The sum of a finite geometric series is the total of all terms in a geometric sequence. It can be calculated by using the formula:

\[ S_n = a_1 \times \frac{1 - r^n}{1 - r} \]
where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula works only when \( r \) is not equal to 1.

In our exercise, the sum of the first 10 terms of the series \( 8, -2, 0.5, -0.125, ... \) is found by plugging the values into the formula, yielding:\[ S_{10} = 8 \times \frac{1 - (-\dfrac{1}{4})^{10}}{1 - (-\dfrac{1}{4})} \].
Understanding how to manipulate this formula is critical for discovering the sum of a series without having to add each term individually, which can be impractical for sequences with a large number of terms.

Arithmetic and Geometric Series

Arithmetic and geometric series are two fundamental types of series in mathematics. Each has distinct characteristics and formulas used to determine the sum of their respective sequences. An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence in which each term after the first is obtained by adding a constant difference to the previous term. The formula for the sum of the first n terms in an arithmetic series is:\[ S_n = \dfrac{n}{2}(a_1 + a_n) \],where \( a_1 \) is the first term, and \( a_n \) is the nth term.

Unlike the arithmetic series where the difference is constant, a geometric series, as described earlier, involves a constant ratio. For the geometric series, it's crucial to note that the sum formula only applies to finite series. Infinite geometric series have a different formula and may converge to a finite value only if the common ratio's absolute value is less than 1.

While arithmetic series increase or decrease linearly, geometric series grow exponentially (for |r| > 1) or decay exponentially (for |r| < 1). Understanding these differences helps students grasp more complex mathematical concepts and solve related problems with greater ease.