Problem 62 - Algebra 2

Question

CHALLENGE State whether each statement is true or false. Explain your reasoning. Doubling the number of terms in an arithmetic series, but keeping the first term and common difference the same, will double the sum.

Step-by-Step Solution

Answer

The statement is false; the sum more than doubles.

Step 1: Understanding the Problem

We need to determine if the sum of an arithmetic series doubles when the number of terms is doubled while keeping the first term and common difference the same.

Step 2: Arithmetic Series Formula

The sum of the first \( n \) terms of an arithmetic sequence is given by the formula:\[ S_n = \frac{n}{2} (2a + (n-1)d) \]where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

Step 3: Original Series Sum Calculation

Using the sum formula for the original series with \( n \) terms:\[ S_n = \frac{n}{2} (2a + (n-1)d) \].

Step 4: Doubling the Number of Terms

Now, consider a series with \( 2n \) terms. Its sum is:\[ S_{2n} = \frac{2n}{2} \left(2a + (2n-1)d\right) = n \left(2a + (2n-1)d\right) \].

Step 5: Comparing the Sums

Compare \( S_{2n} \) with \( 2S_n \):\[ 2S_n = 2 \times \frac{n}{2} (2a + (n-1)d) = n (2a + (n-1)d) \].\[ S_{2n} = n (2a + (2n-1)d) \].The extra term \( nd \) in \( S_{2n} \) means that the sum doesn't simply double.

Step 6: Conclusion

Doubling the number of terms in an arithmetic series will more than double the sum, because the increase involves additional terms with the common difference \( d \). Therefore, the statement is false.

Key Concepts

Series Sum Formula

An arithmetic series represents the sum of terms in an arithmetic sequence. The arithmetic series sum formula is a fundamental tool used to quickly calculate this sum. For any given arithmetic series with a first term of \(a\), a common difference \(d\), and \(n\) terms, you can find the total sum using the formula: \[ S_n = \frac{n}{2} (2a + (n-1)d) \] This formula gives you the sum of all the terms in the series. Here’s how it works:

• \(n\) represents the number of terms you want to sum up.
• \(a\) is the first term of your series. The starting point of your sequence.
• \(d\) is the common difference, the amount you add to each term to get the next.

By incorporating these variables into the sum formula, you can efficiently find the total sum for any arithmetic series.

Common Difference

The common difference \(d\) is what makes an arithmetic sequence unique. It is a fixed value added to each term in the sequence to generate the subsequent term. Understanding the common difference is crucial because it determines the rate of change between terms. For an arithmetic sequence, this difference is constant:

• If \(a_1\) is your first term, then each subsequent term can be found as \(a_2 = a_1 + d, a_3 = a_2 + d, \) and so on.
• The consistent increase or decrease by this difference is what forms the arithmetic pattern.

Therefore, if you know this common difference, you can predict any term in the sequence, making it simpler to compute the sum with the series sum formula.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant called the common difference \(d\). Recognizing this pattern helps in identifying the structure of the sequence and simplifying computations like predicting future terms or calculating the sum. Why is the arithmetic sequence important?

• It offers a straightforward way to understand progressions in numbers, which can be applied in various mathematical and real-world scenarios.
• Each term in the sequence can be represented as: \( a_n = a_1 + (n-1)d \), where \(a_1\) is the first term.

With this formula, you can find any term within the series, facilitating quick summations and analyses.

Sum of Terms

The sum of terms in an arithmetic sequence is where the magic of the series sum formula shines. Calculating this sum by hand would be cumbersome, but the formula \( S_n = \frac{n}{2} (2a + (n-1)d) \) simplifies this process significantly. Consider an arithmetic series where you want to find the total sum, you will:

• Start by identifying the first term \(a\) and the common difference \(d\).
• Determine how many terms \(n\) you want to sum.
• Plug these values into the formula to get your result swiftly.

In contexts such as the original exercise, using this formula reveals nuanced effects of changes in \(n\), such as the observation that doubling the number of terms doesn’t simply double the sum due to the extra components brought in by the common difference.