Problem 69 - Algebra 2

Question

Evaluate the finite series for the specified number of terms. $$ 2+4+8+\ldots ; n=5 $$

Step-by-Step Solution

Answer

The sum of the first five terms in the series is 62.

Step 1: Identify the First Term (a) and the Common Ratio (r)

In a geometric series, the first term is represented as \(a\) while the common ratio is represented as \(r\). Subsequent terms in the series are generated by multiplying the previous term by this common ratio. For the given series, the first term \(a\) is 2 and the common ratio \(r\) is 2.

Step 2: Apply the Formula for the Sum of Finite Geometric Series

The sum \(S\) of the first \(n\) terms of a geometric series can be calculated using the formula: \(S = a \times \frac{1 - r^n}{1 - r}\). Substitute the given values \(a = 2\), \(r = 2\), and \(n = 5\) into the formula.

Step 3: Evaluate the Sum

Substituting the values into the formula gives: \(S = 2 \times \frac{1 - 2^5}{1 - 2}\). Evaluating this expression yields \(S = -62\). However, since the denominator is negative, the negative signs cancel out, giving a sum of 62.

Key Concepts

Finite Series

A finite series is a set of terms that have an end. Think of them as a list of numbers where you have a last number in sight. In the context of a geometric series, this means the series has a specific number of terms we need to add together. For example, if a series goes as 2, 4, 8, and so on, up to the 5th term, we would stop after adding the number corresponding to the 5th term.

There is a start and an end, and we can calculate the sum of all these terms using specific rules. This helps when we do not want to add each number one by one but instead want a quick way to find out the total sum of the series.

Why is this handy? A finite series gives us control over our calculations. Since it has a defined number of terms, it's much easier to handle. We know exactly where it starts and stops, making calculations precise and less prone to errors.

Common Ratio

The common ratio in a geometric series is a key part that makes the series "geometric." It's a constant number that we use to multiply each term to find the next term in the series. For example, if the series starts at 2 and each term needs to double, our common ratio is 2.

It’s like a recipe you follow to create each term from the one before it. Knowing the common ratio can help in predicting each future number in the sequence without having to calculate it from scratch every time. It's a simple rule that saves time and effort.

Understanding the common ratio lets us easily construct any number of terms in a series. It answers the question: If I know one number, how do I find the next? This makes planning and calculation straightforward and efficient.

Sum of Series

Calculating the sum of a series, especially a geometric one, requires a bit of strategy. Thankfully, there’s a special formula for this job. When dealing with a finite series, the sum formula helps us add up all the terms without individually counting each one.

The formula used is: \[ S = a \times \frac{1 - r^n}{1 - r} \]Where:

• \( S \) is the total sum of the series.
• \( a \) is the first term in the series.
• \( r \) is the common ratio.
• \( n \) is the number of terms we are adding.

We plug our known values into this formula to get the sum quickly. For example, with a first term of 2, a common ratio of 2, and 5 terms, it provides the answer in a snap.

Using the formula is like using a powerful tool that manages many little sums at once. It cuts down the work and gives us the final answer efficiently. This way, we focus more on understanding the pattern of the series than worrying about each individual term's addition.