Question

MAKING AN ARGUMENT Your friend believes the sum of a series doubles when the common difference of an arithmetic series is doubled and the first term and number of terms in the series remain unchanged. Is your friend correct? Explain your reasoning.

Short answer

No, the friend is incorrect. Doubling the common difference of an arithmetic series does not double the sum of the series. Even though the common difference is doubled, the sum of the series is not, because the increase of each term in the series is not uniformly distributed but increases as the series progresses.

Step-by-step solution

Understand the formula for the sum of an arithmetic series.

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

Substituting the values into the formula for the original series.

Substitute the values of a, n, and d into the formula \(S = (n/2)[2a + (n-1)d]\). The sum of the series will be denoted as S1.

Substituting the values into the formula for the series with the doubled difference.

Now double the common difference (2d) keeping the first term (a) and number of terms (n) unchanged. Substitute these values into the formula again to find the sum of the new series. This will be denoted as S2.

Analyze whether the sum of the series doubles.

Compare S1 and S2 to determine whether S2 is double of S1. This will answer the given inquiry.

Key concepts

Sum of a Series

When we talk about the sum of a series in arithmetic terms, we refer to finding the total when adding all terms of an arithmetic sequence. An arithmetic sequence consists of numbers arranged in a definite pattern with a common difference between consecutive terms. The formula to find the sum of an arithmetic series is crucial and is given by \[ S = \frac{n}{2} [2a + (n-1)d] \] where:

• \(S\) is the sum of the series.
• \(n\) is the number of terms.
• \(a\) is the first term in the series.
• \(d\) is the common difference between the terms.

Understanding this formula allows students to calculate the sum effectively by plugging in known values and solving. It ultimately helps in realizing the changes in the series when any of these components is altered.

Common Difference

The concept of the common difference is central to arithmetic sequences and directly determines the progression from one term to the next. In simple terms, the common difference \(d\) is the fixed amount added to each term in the sequence to obtain the next term. Let's see how doubling the common difference impacts the series. In our exercise, we have:

• Original common difference: \(d\)
• Doubled common difference: \(2d\)

By doubling \(d\), the relationship between consecutive terms changes, but the alteration can have unexpected effects on the sum of the series, as opposed to a straightforward doubling effect. This forms the basis for analyzing your friend's claim.

Arithmetic Sequence Formula

The arithmetic sequence formula is a vital tool for understanding and solving arithmetic series problems. This formula helps us identify terms in the sequence and sum them up effectively. The individual term in an arithmetic sequence can be calculated as: \[ a_n = a + (n-1)d \] where:

• \(a_n\) is the \(n\)-th term of the sequence.
• \(a\) is the first term.
• \(d\) is the common difference.

Using this formula within the larger sum formula, it shows how the arithmetic series is structured and how each term contributes to the total sum. In the discussed exercise scenario, it plays a key role in exploring what happens to the sequence and its total when the common difference is doubled.

Reasoning in Mathematics

Reasoning in mathematics involves creating arguments and making sense of mathematical concepts to draw valid conclusions. In addressing your friend's belief, reasoning is used to analyze and apply the formula for the sum of an arithmetic series, involving critical thinking. Here's how reasoning applies here:

• Understand each element of the arithmetic sequence formula and how it contributes to the sum.
• Apply changes to the common difference (doubling it) and observe the mathematical consequences, highlighting they don't always result in anticipated outcomes.
• Critically evaluate the outcome of \(S_1\) and \(S_2\) to determine if your friend's hypothesis holds true.

By engaging in this step-by-step reasoning, one builds a comprehensive understanding of arithmetic series, not just viewing it through the lens of simple arithmetic operations, but appreciating the logic and structures that govern these mathematical processes.