Problem 22 - Algebra 2

Question

Find \(S_{n}\) for each geometric series described. $$ a_{1}=4, r=-3, n=5 $$

Step-by-Step Solution

Answer

The sum is 244.

Step 1: Identify the formula

To find the sum of the first n terms of a geometric series, use the formula:\[ S_n = a_1 \frac{1-r^n}{1-r} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 2: Substitute known values into the formula

Given \( a_1 = 4 \), \( r = -3 \), and \( n = 5 \), substitute these into the formula:\[ S_5 = 4 \frac{1-(-3)^5}{1-(-3)} \]

Step 3: Calculate \( r^n \) and simplify the numerator

Calculate \((-3)^5\):\[ (-3)^5 = -243 \]Substitute back into the formula:\[ S_5 = 4 \frac{1 - (-243)}{1+3} \]Simplify the numerator:\[ 1 - (-243) = 1 + 243 = 244 \]

Step 4: Simplify the denominator

Calculate the denominator:\[ 1+3 = 4 \]

Step 5: Calculate the sum

Substitute the simplified numerator and denominator back:\[ S_5 = 4 \frac{244}{4} \]Perform the division:\[ S_5 = 4 imes 61 = 244 \]

Step 6: Write the final answer

Therefore, the sum of the first 5 terms of the geometric series is \( 244 \).

Key Concepts

Sum of a Geometric Series

The sum of a geometric series is an important concept in mathematics, as it allows you to calculate the total of all terms in the series. A geometric series is a series in which each term is found by multiplying the previous term by a constant called the common ratio.
The formula to find the sum of the first n terms of a geometric series is given by:\[ S_n = a_1 \frac{1-r^n}{1-r} \]Here:

• \( S_n \) represents the sum of the first n terms.
• \( a_1 \) is the first term of the series.
• \( r \) is the common ratio.
• \( n \) is the number of terms.

By using this formula, you can find the cumulative value of the first few terms of a sequence without having to manually add each term. This saves both time and effort when dealing with a geometric series that has many terms.

First Term

In any geometric series, the first term, labeled as \( a_1 \), is crucial because it serves as the starting point of the series. It is this initial number that each subsequent term is derived from by repeatedly multiplying by the common ratio.In our given example, the first term is \( a_1 = 4 \). This means that our series starts at 4, and every other term in the series is calculated based on this very first ingredient.
Understanding the first term is the foundation for finding the entire sum of the series, as it is used directly in the summation formula for geometric sequences.

Common Ratio

The common ratio is a fixed number that each term of the geometric series is multiplied by to get the next term. It is represented by \( r \) in the formula.For our specific series, the common ratio is \( r = -3 \). This means each term is multiplied by -3 to obtain the following term. The common ratio can significantly affect the series depending on whether it is positive, negative, greater than or less than one.
When calculating the sum, the common ratio helps determine how quickly the series grows or shrinks, and thus, it plays a vital role in finding \( S_n \) using the geometric series formula.

Number of Terms

The number of terms, represented as \( n \) in the formula, signifies how many terms you want to include in your summation. Knowing \( n \) is necessary for correctly applying the geometric series formula as it defines the boundary of your summation.In the example provided, \( n = 5 \). This means we are summing up the first 5 terms starting from the first term and applying the common ratio repeatedly.
Determining the number of terms in the series is essential for accurate computation and ensures you find the correct sum of the specific segment of your geometric series that you're interested in.