Problem 2 - Applied Calculus

Question

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula. $$ 50+50(0.9)+50(0.9)^{2}+50(0.9)^{3} $$

Step-by-Step Solution

Answer

The sum of the series is 171.95.

Step 1: Understanding the Series

The series given is \( 50 + 50(0.9) + 50(0.9)^2 + 50(0.9)^3 \). It consists of four terms, each starting with 50 and the factor of 0.9 increasing in powers. It looks like a geometric series with the first term as 50 and the common ratio 0.9.

Step 2: Adding Terms Directly

Let's add each term individually: \( 50 \), \( 50 \times 0.9 = 45 \), \( 50 \times 0.9^2 = 40.5 \), and \( 50 \times 0.9^3 = 36.45 \). Adding these gives: \( 50 + 45 + 40.5 + 36.45 = 171.95 \).

Step 3: Identify the Geometric Series Formula

The sum of the first \( n \) terms in a geometric series can be calculated using the formula: \( S_n = a \frac{1-r^n}{1-r} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 4: Apply the Geometric Series Formula

Here, \( a = 50 \), \( r = 0.9 \), and \( n = 4 \). Plug these into the formula: \[ S_4 = 50 \frac{1 - 0.9^4}{1 - 0.9} \]. Calculating further, \( 0.9^4 = 0.6561 \). Thus, \[ S_4 = 50 \frac{1 - 0.6561}{0.1} = 50 \times 3.4389 = 171.945 \].

Step 5: Compare Both Methods

Both ways, whether adding terms directly or using the geometric series formula, we reach the same conclusion for the sum which is approximately \( 171.95 \).

Key Concepts

Common Ratio

In the context of a geometric series, the common ratio is a crucial element. It is the fixed factor by which each term of the series is multiplied to yield the next term. This characteristic gives geometric series their distinctive multiplicative structure.

For instance, in the series given: \(50 + 50(0.9) + 50(0.9)^{2} + 50(0.9)^{3}\), the common ratio \(r\) is \(0.9\). It signifies that to find each successive term, we multiply the previous term by \(0.9\).

Remember, the common ratio can be calculated by dividing any term in the series by its preceding term. In our series:

• \(\frac{50(0.9)}{50} = 0.9\)
• \(\frac{50(0.9)^{2}}{50(0.9)} = 0.9\)
• \(\frac{50(0.9)^{3}}{50(0.9)^{2}} = 0.9\)

This illustrates that the common ratio remains constant throughout a geometric series.

Geometric Series Formula

The geometric series formula is a tool that allows us to calculate the sum of a finite number of terms in a geometric sequence efficiently.

The formula for the sum \(S_n\) of the first \(n\) terms of a geometric series can be expressed as:

• \(S_n = a \frac{1-r^n}{1-r}\)

where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. This formula stems from the pattern of multiplying each term by the common ratio, \(r\), and systematically builds the series.

In the example series, the parameters are:

• \(a = 50\)
• \(r = 0.9\)
• \(n = 4\)

Plugging these into the formula gives us:\[S_4 = 50 \frac{1 - 0.9^4}{1 - 0.9}\]Calculating this provides an accurate total of the series’ sum without having to manually add each term.

Sum of Series

Calculating the sum of a series, especially a geometric series, is an essential task in various mathematical and real-world applications.

For a geometric series, there's a straightforward approach to finding the sum via direct addition of terms or by using the geometric series formula.

For our example: - To find the sum of \(50 + 50(0.9) + 50(0.9)^2 + 50(0.9)^3\) by direct addition, we calculate each term:

• \(50\)
• \(50 \times 0.9 = 45\)
• \(50 \times 0.9^2 = 40.5\)
• \(50 \times 0.9^3 = 36.45\)

Summing these terms results in \(171.95\).- Alternatively, using the geometric series formula, we apply: \[S_4 = 50 \frac{1 - 0.6561}{0.1}\] This calculation also results in approximately \(171.95\).Both methods effectively yield the same sum, but the formula is efficient for longer series.