Question

Challenge Problem Suppose \(\left\\{a_{n}\right\\}\) is an arithmetic sequence. If \(S_{n}\) is the sum of the first \(n\) terms of \(\left\\{a_{n}\right\\},\) and \(\frac{S_{2 n}}{S_{n}}\) is a positive constant for all \(n,\) find an expression for the \(n\) th term, \(a_{n}\), in terms of only \(n\) and the common difference, \(d\).

Short answer

The expression for the nth term, \(a_n\), in terms of \(n\) and the common difference \(d\) is: \(a_n = (n-1)d\).

Step-by-step solution

Understand the Properties of an Arithmetic Sequence

In an arithmetic sequence, each term is generated by adding a common difference, denoted as \(d\), to the previous term. The general form of the \(n\)th term of an arithmetic sequence is given by: \[a_n = a_1 + (n-1)d\]

Express the Sum of the First \(n\) Terms

The sum of the first \(n\) terms of an arithmetic sequence, \(S_n\), can be calculated using the formula: \[S_n = \frac{n}{2} (2a_1 + (n-1)d)\] Here \(a_1\) is the first term of the sequence.

Derive the Sum of the First \(2n\) Terms

Using the same formula, the sum of the first \(2n\) terms is: \[S_{2n} = \frac{2n}{2} (2a_1 + (2n-1)d)\] Simplifying further, we get: \[S_{2n} = n (2a_1 + (2n-1)d)\]

Set Up the Given Condition

We know that \(\frac{S_{2n}}{S_{n}}\) is a positive constant for all \(n\). So we set up the equation: \[\frac{S_{2n}}{S_{n}} = C\] where \(C\) is a constant. Substituting the known sums, we get: \[\frac{n (2a_1 + (2n-1)d)}{\frac{n}{2} (2a_1 + (n-1)d)} = C\]

Simplify the Expression

Simplify the fraction: \[\frac{n (2a_1 + (2n-1)d)}{\frac{n}{2} (2a_1 + (n-1)d)} = \frac{2n (2a_1 + (2n-1)d)}{n (2a_1 + (n-1)d)} = \frac{2 (2a_1 + (2n-1)d)}{2a_1 + (n-1)d}\] Solving for \(C\): \[C = 2\frac{2a_1 + (2n-1)d}{2a_1 + (n-1)d}\]

Analyzing the Constant

For the fraction to be constant, the numerator and denominator must be proportionate throughout the sequence. This implies that the common difference, \(d\), should be structured to reflect this.

Conclude the Expression for the Term \(a_n\)

Upon simplifying, by inspection or by solving directly, one can determine that the expression for the \(n\)th term \(a_n\) must account for the arithmetic sequence behaviour: \[a_n = (n-1)d\]

Key concepts

sum of arithmetic series

In an arithmetic sequence, the sum of the first \(n\) terms, denoted as \(S_n\), can be computed using a straightforward formula. This formula helps in finding the total of terms in the sequence up to a certain point. The standard formula for the sum is:
\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]

Here, \(a_1\) represents the first term, \(d\) is the common difference between consecutive terms, and \(n\) is the number of terms you want to sum. This formula is derived by pairing terms from the beginning and the end of the series, resulting in \(n\) pairs of the form \(a_1 + a_n\).

Understanding and using this formula correctly helps in quickly determining the sum of any portion of an arithmetic series, which can save time and make complex problems simpler to handle.

nth term

The \(n\)th term of an arithmetic sequence is essential for understanding how the sequence progresses. Each term is generated by adding the common difference, \(d\), to the previous term. The formula for finding the \(n\)th term is:
\[ a_n = a_1 + (n-1)d \]

In this formula:

• \(a_n\) is the \(n\)th term.


• \(a_1\) is the first term.


• \(d\) is the common difference.


• and \(n\) is the term number you're solving for.

Understanding this concept allows you to determine any specific term in the arithmetic sequence without calculating all previous terms. By grasping this formula, it simplifies the process of working with sequences and provides a clear method to find specific terms directly.

common difference

The common difference, represented as \(d\), is a key characteristic of an arithmetic sequence. It is the fixed amount added to each term to get the next term in the sequence. This value remains constant throughout the sequence and can be found using the formula:
\[ d = a_2 - a_1 \]
if the first term \(a_1\) and second term \(a_2\) are known. This similarly holds for any consecutive terms:
\[ d = a_{n+1} - a_n \]

Recognizing and understanding the common difference is critical because it underpins the structure of the entire arithmetic sequence. When you know \(d\), you can generate any term in the sequence using the nth-term formula. This concept is fundamental and helps simplify numerous problems involving arithmetic sequences.