Problem 47 - Algebra 2

Question

The sum of an infinite geometric series is \(125,\) and the value of \(r\) is 0.4. Find the first three terms of the series.

Step-by-Step Solution

Answer

The first three terms are 75, 30, and 12.

Step 1: Understanding the formula for the sum of an infinite geometric series

The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio. In this problem, \( S = 125 \) and \( r = 0.4 \).

Step 2: Solving for the first term \( a \)

Substitute the known values into the formula \( S = \frac{a}{1 - r} \). We have \( 125 = \frac{a}{1 - 0.4} \). Simplifying, we obtain \( 125 = \frac{a}{0.6} \). To find \( a \), solve \( a = 125 \times 0.6 \) which gives \( a = 75 \).

Step 3: Calculating the first term

The first term \( a \) of the series is therefore \( 75 \).

Step 4: Calculating the second term

To find the second term, use the formula for the nth term of a geometric series: \( a_n = ar^{n-1} \). Substitute \( n = 2 \), \( a = 75 \), and \( r = 0.4 \), we get \( a_2 = 75 \times 0.4 = 30 \).

Step 5: Calculating the third term

For the third term, use the same formula. Substitute \( n = 3 \), \( a = 75 \), and \( r = 0.4 \), we get \( a_3 = 75 \times (0.4)^2 = 75 \times 0.16 = 12 \).

Step 6: Summarizing the first three terms

The first three terms of the series are \( 75 \), \( 30 \), and \( 12 \).

Key Concepts

sum of a series

When we talk about the sum of a series, we are referring to the total value you get when adding all the terms of a sequence. For an infinite geometric series, like the one in our problem, we use a special formula to find this sum. The key is that the series must converge, which means the terms get closer and closer to a specific value as you move further in the sequence.

The formula used for the sum of an infinite geometric series is given by: \[ S = \frac{a}{1 - r} \] Where:

• \( S \) is the sum of the series.
• \( a \) is the first term of the series.
• \( r \) is the common ratio between terms.

In our example, the sum \( S \) is 125, making it a critical part of calculating other terms of the series. By solving this formula for \( a \), we discovered that the first term of the series was 75.

geometric sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This kind of sequence is predictable and systematic, making it very useful in mathematics.

In the case of our infinite geometric series, we know:

• \( a \), the first term, is 75.
• \( r \), the common ratio, is 0.4.

Thus, each new term in the sequence is just the previous term multiplied by 0.4. The sequence starts at 75, with the subsequent terms being calculated by multiples of 0.4 such as the second term (30) and third term (12), which show how quickly the terms decrease in value.

common ratio

The common ratio is a key characteristic in geometric sequences and series. It is the constant factor between consecutive terms of a geometric sequence. In our problem, the common ratio \( r \) is 0.4. This ratio determines how fast the terms in the series shrink or grow.

To understand why it is called the "common" ratio:- It is "common" because it is the same throughout the series.- It simplifies the process of finding subsequent terms because you only need to multiply the previous term by \( r \).
In summary, knowing the common ratio tells you a lot about the behavior of the series. A common ratio less than 1, as in this case with 0.4, indicates the series will converge, meaning its terms will decrease and approach a total, finite sum as opposed to diverging and reaching infinity.