Question

How many terms are there in the series Find the sum. $$ \sum_{k=2}^{20}(-1)^{k} 4(0.8)^{k} $$

Short answer

Sum: $$ \sum_{k=2}^{20}(-1)^{k} 4(0.8)^{k} $$ Answer: There are 19 terms in the series, and the sum of the series is approximately $$ \frac{2.56(1 - (-0.8)^{19})}{1.8}. $$

Step-by-step solution

Determine the series terms

In the given sum, $$ \sum_{k=2}^{20}(-1)^{k} 4(0.8)^{k}, $$ we have the first term with \(k = 2\) and the last term with \(k = 20\).

Calculate the number of terms

To find the number of terms in the series, subtract the first term index from the last term index, and add 1 (to include both the first and last terms): $$ \text{Number of terms} = (20 - 2) + 1 = 19. $$

Identify this as a geometric series

The given series is a geometric series, as there is a common ratio between consecutive terms. The common ratio is given by $$ (-1)(0.8) = - 0.8. $$

Find the first term of the series

Plug in the value of the first term index (\(k=2\)) into the series expression: $$ a = (-1)^2 4(0.8)^2 = 4(0.64) = 2.56. $$

Apply the formula for the sum of a geometric series

For a geometric series with a first term \(a\), common ratio \(r\), and number of terms \(n\), the sum is given by: $$ S = \frac{a (1 - r^n) }{1 - r}. $$

Find the sum of the series

Plug in the values for \(a\), \(r\), and \(n\) to find the sum: $$ S = \frac{2.56 (1 - (-0.8)^{19})}{1 - (-0.8)} = \frac{2.56(1 - (-0.8)^{19})}{1.8}. $$ Calculate the numerical value of the sum. Thus, there are 19 terms in the series, and the sum of the series can be calculated using the formula for the sum of a geometric series.

Key concepts

Sum of a Series

The sum of a series is a way to combine all the individual elements of the set into a single value. In a geometric series, this process becomes more systematic given its predictable nature. A geometric series is characterized by each term being a constant multiple, known as the common ratio, of the term preceding it.

The formula to find the sum of a geometric series is:

• First term of the series: \( a \)
• Common ratio: \( r \)
• Number of terms: \( n \)

The sum, \( S \), is given by the formula:\[S = \frac{a(1 - r^n)}{1 - r}\]For a geometric series, after calculating the first term, using the common ratio, and knowing the number of terms, you can efficiently determine the series' sum. Always ensure that the common ratio is not equal to 1, as this formula does not apply if the series is constant.

Number of Terms in a Series

Finding the number of terms in a series is straightforward once you understand the indexing of the sequence. In any given series, each term is typically represented by an index value. The index tells us the position of each term within the sequence.

To determine how many terms are present, use the following simple formula:

• Identify the index of the first term.
• Identify the index of the last term.
• Subtract the index of the first term from the index of the last term.
• Add one to include both the first and last term in the count.

Mathematically, if the index of the first term is \( k_1 \) and the index of the last term is \( k_2 \), the number of terms \( n \) is given by:\[n = (k_2 - k_1) + 1\]This method ensures that no terms are missed in your count, accurately describing the sequence length.

Common Ratio

In a geometric series, the common ratio is a vital component because it defines the pattern of progression through the series. This ratio is the factor by which each term increases or decreases to arrive at the next term.

To find the common ratio, \( r \), in a geometric series:

• Take any term in the series (except the first) and divide it by the previous term.

This ratio remains consistent throughout the series, which is why the pattern in a geometric series is uniform. Let's consider an example to clarify:

• If the first term is 4 and the second term is -3.2, the common ratio \( r \) is calculated by dividing the second term by the first.
• So, \((-3.2 / 4) = -0.8\), thus, the common ratio is \(-0.8\).

The common ratio provides insights into the series behavior, such as whether the series is converging, diverging, or oscillating.