Problem 38 - Algebra 2

Question

Find the sum of each geometric series to the given term. $$ 20+(-2)+0.2+\ldots ; n=12 $$

Step-by-Step Solution

Answer

The sum of the first 12 terms of the given geometric series is approximately 16.50.

Step 1: Identify the first term and the common ratio

In a geometric series, the first term (a) is clearly given as 20. The common ratio (r) can be found by dividing the second term by the first term, which gives \(r = \frac{-2}{20} = -0.1\).

Step 2: Substitute into the formula

The formula for the sum of the first \(n\) terms of a geometric series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). Substituting \(a = 20\), \(r = -0.1\), and \(n = 12\) into this formula, we get \(S_{12} = \frac{20(1 - (-0.1)^{12})}{1 - (-0.1)}\).

Step 3: Simplify the equation

Solving the equation gives \[S_{12} = \frac{20(1 - (-0.1)^{12})}{1.1} \approx 16.5029.\]

Step 4: Round off to desired accuracy

Rounding off to two decimal places for more precision, the sum of the terms of the series up to the 12th term is approximately 16.50.

Key Concepts

Common Ratio

In a geometric series, the concept of the common ratio is vital. It is a constant factor that connects consecutive terms in a series. This means that each term can be found by multiplying the previous term by this common ratio.
For the series given in the exercise, the first two terms are 20 and -2. To find the common ratio, you divide the second term by the first term, like so:

• Second term: -2
• First term: 20
• Common Ratio: \( r = \frac{-2}{20} = -0.1 \)

This negative ratio indicates an alternation of sign in the sequence as the series progresses. Recognizing the common ratio is essential to work out further terms and sums of the geometric series effectively.

Sum of a Geometric Series

Finding the sum of a geometric series is a common task in mathematics, especially when dealing with sequences where each term builds upon the previous one. The formula for the sum of the first \( n \) terms of a geometric progression is:
\[S_n = \frac{a(1 - r^n)}{1 - r}\]
Here:

• \( S_n \) is the sum of the first \( n \) terms.
• \( a \) is the first term of the series.
• \( r \) is the common ratio.
• \( n \) is the term number up to which the sum is calculated.

In the exercise, substituting \(a = 20\), \(r = -0.1\), and \(n = 12\) allows you to use this formula:
\[S_{12} = \frac{20(1 - (-0.1)^{12})}{1 - (-0.1)}\]
After simplifying, you will find \(S_{12}\) to be approximately 16.50 for the given series. Using this formula efficiently calculates the total of any geometric series to a specified number of terms.

Geometric Progression

A geometric progression is a type of sequence where each term after the first one is derived by multiplying the previous term by a fixed, non-zero number called the common ratio. This progression is different from arithmetic sequences, where terms are generated by adding a constant value.
In our given series, starting with 20, each subsequent term is obtained from the previous one by multiplying by the common ratio \(-0.1\). Hence the sequence looks like:

• 20
• -2 (20 multiplied by -0.1)
• 0.2 (-2 multiplied by -0.1)
• ... (continuing this pattern)

The characteristics of a geometric progression allow for a clear and structured approach to find any term in the progression or the cumulative sum up to a specified point. Recognizing these sequences makes it easier to predict future terms given the first term and the common ratio.