Problem 61 - Algebra 2

Question

CHALLENGE State whether each statement is true or false. Explain your reasoning. Doubling each term in an arithmetic series will double the sum.

Step-by-Step Solution

Answer

True; doubling each term in an arithmetic series doubles the sum.

Step 1: Understand the Definition of an Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term is derived by adding a fixed, constant difference to the previous term. For an arithmetic series with the first term \(a\), common difference \(d\), and \(n\) terms, the sum \(S_n\) is given by: \[ S_n = \frac{n}{2} \cdot (2a + (n-1)d) \]

Step 2: Consider Doubling Each Term of the Arithmetic Series

If each term in the arithmetic series is doubled, each will become \(2a, 2(a + d), 2(a + 2d), \dots, 2(a + (n-1)d)\). The common difference now becomes \(2d\).

Step 3: Calculate the Sum of the Modified Series

The sum of this new series, where each term is doubled, is given by: \[ S'_n = \frac{n}{2} \cdot [2(2a) + (n-1)(2d)] = \frac{n}{2} \cdot 2[2a + (n-1)d] = 2S_n \] This means doubling each term doubles the sum of the series.

Step 4: Verify with a Simple Example

Consider a simple arithmetic series: 2 + 4 + 6. The sum is 12. Doubling each term gives: 4 + 8 + 12, and the sum is 24. Since 24 is double 12, the statement holds for this example.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers in which each term after the first is produced by adding a constant to the previous term. This constant is known as the common difference.
An example will help to clarify: imagine you have a sequence 2, 4, 6, 8. Here, the common difference is 2, because you add 2 to each existing term to get to the next one.

• The first term is called the initial term, denoted as \(a\).
• The common difference is denoted as \(d\).
• The formula to find any term \(a_n\) in an arithmetic sequence is: \(a_n = a + (n-1)d\)

Arithmetic sequences are foundational to understanding arithmetic series, which we'll discuss next.

Sum of Series

The sum of an arithmetic series refers to the sum of all terms in an arithmetic sequence. Whether it involves adding a few terms or many, the concept remains the same.
To find the sum \(S_n\) of an arithmetic series, use the formula: \[ S_n = \frac{n}{2} \, (2a + (n-1)d) \]
Here, \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference. This formula works by calculating the average of the first and last terms and then multiplying by the number of terms.

• The factor of \(\frac{n}{2}\) accounts for the pairing of terms from the beginning and end.
• When you sum an arithmetic series, every adjustment in terms reflects linearly thanks to the arithmetic nature.

As such, doubling every term in an arithmetic series doubles its sum, which was verified in the solution.

Common Difference

The common difference, denoted as \(d\), is a crucial element in arithmetic sequences and series. It defines the constant step from one term to the next in the sequence.
Calculating the common difference is simply subtracting any term from the subsequent one. For example, in the sequence 3, 6, 9, 12, the common difference is 3 because \(6 - 3 = 3\).

• The common difference reveals the uniformity of increase or decrease between sequence terms.
• If you alter the sequence by doubling all terms, the common difference will also double, as shown by how each term itself doubles.
• Understanding \(d\) matters deeply for computations involving sums of series, as it influences both the sequence's formation and its summed result.

This consistency in spacing helps predict future values and understand series behaviors, such as the effect deduced in the challenge problem.