Problem 57 - Algebra 2

Question

Use a graphing calculator to find the sum of each geometric series. $$ \sum_{n=1}^{10} 5(0.2)^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is approximately 6.25.

Step 1: Identify Components of the Geometric Series

First, recognize the format of a geometric series. This series is given by the formula \( \sum_{n=1}^{10} a r^{n-1} \), where \( a \) is the first term, and \( r \) is the common ratio. For this series, \( a = 5 \) and \( r = 0.2 \).

Step 2: Use Geometric Series Sum Formula

The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated with the formula \( S_n = a \frac{1-r^n}{1-r} \). Substitute \( a = 5 \), \( r = 0.2 \), and \( n = 10 \) into the formula.

Step 3: Calculate the Expression Inside the Sum Formula

Calculate \( 0.2^{10} \) using your graphing calculator. This step is needed for the sum formula. Compute \( 0.2^{10} = 0.0000001024 \).

Step 4: Substitute Values into the Formula

Substitute the values into the geometric series sum formula: \( S_{10} = 5 \frac{1 - 0.0000001024}{1 - 0.2} \). Simplify \( 1 - 0.0000001024 = 0.9999998976 \) and \( 1 - 0.2 = 0.8 \).

Step 5: Solve for the Sum

Divide the numerator by the denominator: \( \frac{0.9999998976}{0.8} \approx 1.249999872 \). Multiply by 5 to find the sum: \( S_{10} = 5 \times 1.249999872 \approx 6.25 \).

Step 6: Verify Result Using the Graphing Calculator

Enter the series \( \sum_{n=1}^{10} 5(0.2)^{n-1} \) directly into the summation function of a graphing calculator to confirm the result. This is a good practice to ensure correctness.

Key Concepts

sum formula

In a geometric series, the sum of the series can be calculated without adding each term individually. We have a convenient formula called the "sum formula". For the first \( n \) terms of a geometric series, the sum \( S_n \) can be computed as follows:\[S_n = a \frac{1 - r^n}{1 - r}\]Here, \( a \) represents the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula is especially useful because it saves time and effort, allowing us to quickly find the sum of many terms in a series without performing repetitive calculations.

• First, determine what \( a \), \( r \), and \( n \) are in your specific series.
• Substitute these values into the formula.
• Calculate the power \( r^n \), which may involve using a calculator if \( n \) is large.
• Complete the operations in the numerator and denominator.
• Finally, multiply \( a \) by the result of the division.

After substitution and calculation, the formula provides you with the sum in a streamlined way.

graphing calculator

A graphing calculator can be a powerful tool when working with geometric series. It helps in verifying the sums calculated using formulas and performs operations that might be cumbersome by hand. Here's how it can be particularly helpful:

• Graphing calculators can calculate powers efficiently, which is crucial for evaluating \( r^n \) in the sum formula.
• They provide a summation function, allowing users to input the series directly and obtain the sum. This function automates the entire process.
• Graphing calculators are also great for performing quick checks on intermediate values, such as \( r^n \) or the division in the sum formula to confirm accuracy.

For example, once you compute the sum of the geometric series using the formula, you can enter the series \( \sum_{n=1}^{10} 5(0.2)^{n-1} \) directly into the calculator to ensure your hand calculation is correct. This dual approach—using both manual calculation and technology—provides confidence in the results obtained.

common ratio

In the context of a geometric series, the "common ratio" is a fundamental element as it defines the progression pattern between the terms. The common ratio, \( r \), is the factor by which each term is multiplied to get to the next term in the series.

• If \( r > 1 \), the series will grow larger with each subsequent term.
• If \( 0 < r < 1 \), the terms will decrease, converging towards zero.
• If \( r < 0 \), the terms alternate between positive and negative, with the absolute value either growing or shrinking.

The common ratio is identified by looking at the formula for each term of the geometric series, excluding the first term, it appears as \( r^{n-1} \). It determines the behavior of the sum as more terms are added, thereby influencing the final result significantly. Therefore, correctly identifying \( r \) is crucial in applying the sum formula effectively and understanding the overall pattern of the series.