Problem 64 - Algebra 2

Question

ACT/SAT The first term of a geometric series is \(-1,\) and the common ratio is \(-3 .\) How many terms are in the series if its sum is 182\(?\) $$ \begin{array}{l}{\text { A } 6} \\ {\text { B } 7} \\ {\text { C } 8} \\\ {\text { D } 9}\end{array} $$

Step-by-Step Solution

Answer

The series has 6 terms, choice A is correct.

Step 1: Understanding the Geometric Series Formula

A geometric series is given by 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. In this problem, the first term \( a = -1 \), the common ratio \( r = -3 \), and the sum \( S_n = 182 \). We need to find \( n \).

Step 2: Substituting Known Values

Substitute the given values into the geometric series formula: \( 182 = -1 \frac{1 - (-3)^n}{1 + 3} \).Simplifying the denominator gives: \( 1 + 3 = 4 \), so the formula becomes \( 182 = -1 \frac{1 - (-3)^n}{4} \).

Step 3: Solving for the Numerator

Multiply both sides by 4 to clear the denominator: \( 182 \times 4 = -1 (1 - (-3)^n) \).This simplifies to \( 728 = -1 + (-1)(-3)^n \), or \( 729 = (-3)^n \).

Step 4: Finding the Common Ratio Power

Evaluate \( (-3)^n = 729 \) by recognizing that \( 729 = 3^6 \).Since the power only affects the coefficient and not the sign when \( n \) is even, check if \((-3)^n \) will equal 729 when \( n \) is an odd number, because for negative bases, \((-3)^6 = 729 \) holds: \( n = 6 \).

Step 5: Confirming the Number of Terms

Since \( (-3)^n = 729 \) when \( n = 6 \) (odds yield a negative power result), and calculating confirms a match with 729, hence the number of terms in the series is 6.

Key Concepts

Geometric Sequence

A geometric sequence is a set of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding geometric sequences is crucial because they frequently appear in various mathematical problems and real-world situations.
Here’s how it works:

• First Term (\(a\)): The initial number of the sequence. In our example, this is \(-1\).
• Common Ratio (\(r\)): Each term is derived by multiplying the previous term by this fixed value, which in the exercise is \(-3\).

Each term of the sequence is defined as \(a, ar, ar^2, ar^3, \ldots \). It creates an exponential growth (or decay, if \(r\) is a fraction) pattern, which is why understanding this sequence is essential for tackling problems involving geometric series.

Common Ratio

The common ratio is a key component in both geometric sequences and series. It determines how each term in the sequence relates to the one before it. The importance of the common ratio lies in the way it can dictate the nature of the sequence.
Here are some properties you should know:

• Positive or Negative: The common ratio can be either. In the exercise, it is \(-3\), leading to alternating sign terms.
• Magnitude Influence: A common ratio greater than 1 leads to rapidly increasing values, while between 0 and 1 leads to decreasing values.

Understanding the common ratio helps in identifying the pattern of the sequence and calculating the sum correctly. As seen in the geometric series formula, \(r\) crucially affects the total sum and behavior of the series.

Sum of Series

The sum of a geometric series is often required in algebra problems. The formula to find the sum of the first \(n\) terms of a geometric series is \(S_n = a \frac{1 - r^n}{1 - r}\). This formula’s utility lies in its ability to condense a potentially long sequence into a single computation. Let's break it down:

• First Term (\(a\)): This remains consistent as the initial value of the sequence.
  
• Number of Terms (\(n\)): The number of terms you want to sum up.
• Common Ratio (\(r\)): The multiplier of the sequence.

In this problem, understanding how to manipulate this formula is essential in solving for \(n\) when the sum and common ratio are known, as demonstrated in the solution steps.

Algebraic Problem Solving

Solving algebraic problems involving geometric series requires understanding both the theory behind the series and practical algebraic manipulations. This exercise illustrates how to apply given numerical values into abstract formulas to determine unknown quantities.
Focus on these steps for problem solving:

• Substitute Known Values: Insert the given quantities into the formula to begin isolating the unknowns.
• Simplify Expresssions: This may include multiplying or factoring to eliminate fractions or unnecessary terms.
• Solve for the Unknown: With simplifications made, algebraic techniques such as solving for powers, may be employed.
  

This problem-solving approach was used to determine \(n\), the number of terms, by essentially reverse-engineering the sum formula, highlighted by recognizing powers of numbers, such as 729 is \(3^6\).