Problem 81 - Beginning and Intermediate Algebra

Question

Find the sum of the terms of the infinite geometric sequence, if possible. $$-40,-30,-\frac{45}{2},-\frac{135}{8}, \dots$$

Step-by-Step Solution

Answer

The sum of the terms of the infinite geometric sequence is \(S = -160\).

Step 1: Find the common ratio (r)

Determine the common ratio by dividing the second term by the first term: \(r = \frac{-30}{-40} = \frac{3}{4}\) Now, we need to check if this common ratio holds true for the other terms.

Step 2: Check to see if the common ratio is consistent

Multiply the second term by the common ratio to see if you get the third term: \(-30 * \frac{3}{4} = -\frac{45}{2}\) The third term is indeed equal to \(-\frac{45}{2}\). Now do the same for the 3rd and the 4th term: \(-\frac{45}{2} * \frac{3}{4} = -\frac{135}{8}\) Again, the multiplication works out, so we can confirm that the common ratio for this sequence is \(\frac{3}{4}\).

Step 3: Determine if the series converges

Since the absolute value of the common ratio is less than 1: \(|\frac{3}{4}| < 1\) the series converges.

Step 4: Calculate the sum of the infinite geometric series

Now that we know the series converges, we can use the formula for the sum of an infinite geometric series: \[S = \frac{a}{1-r}\] Plug in the values of the first term a (-40) and the common ratio r (\(\frac{3}{4}\)): \[S = \frac{-40}{1-\frac{3}{4}}\] \[S = \frac{-40}{\frac{1}{4}}\] \[S = -40 * 4\] \[S = -160\] So, the sum of the terms of the infinite geometric sequence is -160.

Key Concepts

Understanding the Common Ratio

The common ratio is a key component in a geometric sequence. In simple terms, it's the number we multiply by each term to get to the next term. Think of it as the factor that links consecutive terms.
To find the common ratio, you divide any term by the one before it. For our sequence

• First Term: -40
• Second Term: -30

The common ratio is \( r = \frac{-30}{-40} = \frac{3}{4} \).
This means every term is \( \frac{3}{4} \) of the one before it.
Checking this consistency is crucial. We verified it with the third and fourth terms:

• \( -30 \times \frac{3}{4} = -\frac{45}{2} \)
• \( -\frac{45}{2} \times \frac{3}{4} = -\frac{135}{8} \)

Seeing the same ratio confirms that our entire sequence adheres to this common factor.

Recognizing a Convergent Series

A geometric series can be infinite, but not all infinite series have a sum that you can find—only those that converge.
The term 'convergent series' means that as you keep adding the terms, they get closer and closer to a specific value.
For convergence in geometric series, the absolute value of the common ratio needs to be less than 1.
In our example, the ratio \( \frac{3}{4} \) satisfies this condition because:

• \( |\frac{3}{4}| < 1 \)

This tells us that the sum of our infinite series settles at a specific number.—Important for ensuring calculations make sense.
Recognizing convergence means we can proceed to find its sum.

Finding the Sum of the Series

Once we know a series converges, we can calculate its sum using a simple formula. For an infinite geometric series, the sum \( S \) is:
\[ S = \frac{a}{1 - r} \]
Where \( a \) is the first term and \( r \) is the common ratio.
Using our values:

• First Term \( a = -40 \)
• Common Ratio \( r = \frac{3}{4} \)

The formula becomes:
\[ S = \frac{-40}{1 - \frac{3}{4}} \]
This simplifies to:
\[ S = \frac{-40}{\frac{1}{4}} \]
Then solving gives:
\[ S = -40 \times 4 = -160 \]
This means the sum of all terms in the infinite sequence is \(-160\).
Understanding and applying this formula is vital to solving infinite series problems.

Exploring the Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by the common ratio.
This type of sequence forms the basis for understanding many mathematical and real-world phenomena.
Let's break down our example sequence:

• First Term: -40
• Second Term: -30
• Third Term: \(-\frac{45}{2}\)
• Fourth Term: \(-\frac{135}{8}\)

Each subsequent term is obtained by multiplying by \( \frac{3}{4} \), the common ratio.
Geometric sequences are powerful for modeling exponential growth or decay, such as in finance or physics.
Grasping this sequence helps us understand various mathematical principles and how they apply in different settings.