Problem 52 - Algebra 2

Question

Determine whether the sum of each infinite geometric series exists. $$ -972-324-108-\dots $$

Step-by-Step Solution

Answer

Therefore, the sum of the infinite geometric series \(-972 - 324 - 108 - \dots\) is -729

Step 1: Identify the first term and the common ratio

In a geometric series, each term is the previous term multiplied by a constant. This constant is the common ratio. In the series \(-972 - 324 - 108 - \dots\), the first term \(a = -972\), and you can determine the common ratio \(r\) by taking any term and dividing it by the previous term. So, \(r = 324 / -972 = -1/3\).

Step 2: Check the range of the ratio

For the sum of an infinite geometric series to exist, the absolute value of the common ratio has to be less than 1, meaning \(-1 < r < 1\). In this case, \(-1/3\) falls within this range, which implies that the sum of this infinite series might exist.

Step 3: Calculate the sum using the formula

With the first term and the common ratio identified, you can use the formula to calculate the sum of the infinite geometric series. The formula is \(S = a / (1 - r)\). Substituting the values we have \(S = -972 / (1 - (-1/3)) = -972 / (4/3) = -729\).

Key Concepts

geometric series sum

An infinite geometric series is a series where each term is created by multiplying the previous term by a constant called the common ratio. In the series given,

• The first term, denoted as \( a \), is \( -972 \).
• The common ratio, \( r \), is \( -\frac{1}{3} \).

To calculate the sum of an infinite geometric series, once you have identified \( a \) and \( r \), you can use the formula:\[S = \frac{a}{1 - r}\]Here, the sum \( S \) can be calculated by substituting \( a = -972 \) and \( r = -\frac{1}{3} \). This gives us:\[S = \frac{-972}{1 - (-\frac{1}{3})} = \frac{-972}{\frac{4}{3}} = -729\]This shows that if the series converges, its sum will be \( -729 \). Hence, recognizing the first term and the common ratio is crucial for understanding and calculating the geometric series sum.

common ratio

The common ratio in a geometric series is a vital element that helps determine the nature of the sequence. It is the constant factor you multiply by to get from one term to the next. For the series

• \(-972, -324, -108, \ldots\)

The common ratio \( r \) is calculated by dividing any term by its preceding term.

• In this series, dividing \( -324 \) by \( -972 \) gives us \( r = -\frac{1}{3} \).

The common ratio tells you whether the series will continue to grow, shrink, or oscillate. If the absolute value of \( r \) is less than 1, the series is generally expected to converge. This concept of common ratio lets us decide quickly if finding the sum is possible, as it influences the convergence of the series directly.

convergence of series

The convergence of a series refers to whether or not the sum of the sequence approaches a finite number. For an infinite geometric series, the critical factor that determines convergence is the common ratio \( r \). The series will converge only if:

• The absolute value of \( r \) is less than 1, i.e., \(-1 < r < 1\).

In our series example with the common ratio \( r = -\frac{1}{3} \), the condition \(-1 < -\frac{1}{3} < 1\) is satisfied. This ensures that the series converges.
Additionally, when a series converges, it implies that the terms of the series will become smaller and smaller in magnitude as we extend further. This pattern implies reaching a specific finite number or limit as the number of terms tends toward infinity.
Understanding convergence helps you know whether it's worthwhile to compute a sum or if the values will simply diverge without landing on a specific result.