Problem 57 - Algebra 2

Question

Find \(S_{n}\) for each geometric series described. \(a_{1}=1, a_{6}=-243, r=-3\)

Step-by-Step Solution

Answer

The sum of the first 6 terms, \(S_6\), is -182.

Step 1: Identify the series parameters

We are given the first term of the geometric series, \(a_1 = 1\), the sixth term, \(a_6 = -243\), and the common ratio, \(r = -3\). Our goal is to find the sum of the first \(n\) terms of this series, \(S_n\).

Step 2: Find the number of terms

Since \(a_6 = -243\), we use the formula for the \(n\)-th term of a geometric series, which is \(a_n = a_1 \cdot r^{n-1}\). Thus, \(-243 = 1 \cdot (-3)^{6-1}\).

Step 3: Solve for the sixth term to verify

Calculate \((-3)^{6-1} = (-3)^5\). This equals \(-243\), confirming that \(a_6 = -243\) matches our series parameters.

Step 4: Use the sum formula for a geometric series

The sum of the first \(n\) terms of a geometric series is given by \(S_n = a_1 \frac{1-r^n}{1-r}\). Substitute \(a_1 = 1\) and \(r = -3\) into this formula. Since we want to find \(S_6\), we set \(n=6\).

Step 5: Calculate the sum \(S_6\)

Substitute into the formula: \[ S_6 = 1 \cdot \frac{1-(-3)^6}{1-(-3)} = \frac{1 - 729}{1 + 3} = \frac{-728}{4} = -182. \]

Step 6: Interpret the result

The calculated result \(S_6 = -182\) represents the sum of the first 6 terms of the given geometric series.

Key Concepts

Common Ratio

In any geometric series, the term "common ratio" is used to describe the factor that each term is multiplied by to get the next term.
This value is crucial because it defines the growth or decay rate of the sequence. If the common ratio, denoted as \( r \), is greater than 1, the series will grow. If it is between 0 and 1, the series will shrink. For negative values, like in our example with \( r = -3 \), the series alternates between positive and negative values.
Calculating the common ratio involves comparing two consecutive terms of the sequence:

• If you know two terms, such as \( a_1 \) and \( a_2 \), find \( r \) by dividing \( a_2 \) by \( a_1 \).
• This verifies if there's consistency across the entire sequence.

Sum of Terms

The "sum of terms" in a geometric series refers to the total of the first \( n \) terms.
This sum can be calculated using a particular formula that depends on the first term, the common ratio, and the number of terms. The formula is:

• \( S_n = a_1 \frac{1-r^n}{1-r} \),

where:

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

To find the sum, substitute the given values and solve the equation. For example, in our exercise, using the formula we've shown how to calculate the sum of the first 6 terms: \( S_6 = -182 \).
This sum provides insight into the behavior of the series over a set period.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant, known as the common ratio. This defines the progression of the terms.
Key characteristics include:

• The first term, \( a_1 \), is given, initiating the sequence.
• Each subsequent term is the product of the previous term and the common ratio \( r \).

For example, in the sequence where \( a_1 = 1 \) and \( r = -3 \):
The terms generated would be 1, -3, 9, -27, etc.
You can easily spot the pattern by continually multiplying by \( -3 \). This consistent pattern allows for simple prediction and calculation of future terms.

Term Formula

The term formula for a geometric series, also known as the formula for the \( n \)-th term, is used to calculate any term in a geometric sequence without listing all the terms.
It is expressed as:

• \( a_n = a_1 \cdot r^{n-1} \),

where:

• \( a_n \) is the \( n \)-th term,
• \( a_1 \) is the first term,
• \( r \) is the common ratio,
• \( n \) denotes the number of terms.

This formula is crucial for verifying the position of any term within the sequence.
In our example, we used \( a_6 = 1 \cdot (-3)^{5} = -243 \).
This shows that the 6th term is indeed \( -243 \). By utilizing this formula, you can quickly determine any term's value without computing all previous terms.