Problem 3 - Precalculus Mathematics for Calculus

Question

True or false? The \(n\) th partial sum of an arithmetic sequence is the average of the first and last terms times \(n .\)

Step-by-Step Solution

Answer

True, the statements are equivalent.

Step 1: Understanding the Arithmetic Sequence

An arithmetic sequence is defined as a sequence of numbers such that the difference between the consecutive terms is constant. This difference is called the common difference, denoted by \( d \). The first term of the sequence is denoted by \( a_1 \), and the \( n \)th term is given by \( a_n = a_1 + (n-1)d \).

Step 2: Understanding the Partial Sum Formula

The \( n \)th partial sum \( S_n \) of an arithmetic sequence is given by the formula \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( a_n \) is the \( n \)th term. Substituting the expression for \( a_n \) gives us \( S_n = \frac{n}{2} (a_1 + a_1 + (n-1)d) = \frac{n}{2} (2a_1 + (n-1)d) \).

Step 3: Average of First and Last Terms

The average of the first and last terms of the sequence is \( \frac{a_1 + a_n}{2} \). Multiplying this average by \( n \) results in \( n \times \frac{(a_1 + a_n)}{2} = \frac{n}{2} (a_1 + a_n) \).

Step 4: Checking Equivalence of Expressions

The expression \( n \times \frac{(a_1 + a_n)}{2} \) derived from considering the average matches with the standard formula for \( S_n \), \( \frac{n}{2} (a_1 + a_n) \). Both expressions result in the same value, verifying the statement.

Key Concepts

Partial Sum

In the context of arithmetic sequences, a partial sum refers to the sum of the first "n" terms in the sequence. Let's break it down to understand it completely.

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant called the "common difference." The partial sum, particularly the nth partial sum, helps us find the total of a specified number of these terms. Instead of adding each individual term one by one, the partial sum formula allows us to calculate this total all at once, which is much faster.

The formula for the nth partial sum is:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

Here, \( S_n \) is the nth partial sum, \( a_1 \) is the first term, and \( a_n \) is the nth term. This formula arises from calculating the average of the first and last terms and then multiplying by the number of terms, "n."

It's an elegant way to efficiently sum up terms without tedious calculations. It’s important to understand all the pieces, so let's keep going with other terms.

Common Difference

The common difference is a crucial concept when dealing with arithmetic sequences. Simply put, it is the consistent amount that each term increases by from one to the next in the sequence.

Mathematically, it is denoted by the letter "d." The common difference can be determined by subtracting any term from the preceding one in the sequence:

• \( d = a_{n+1} - a_n \)

Knowing the common difference allows us to construct or analyze the entire sequence. Once you have "d" and the first term \( a_1 \), you can predict any term in the sequence or calculate sums using the appropriate formulas.

In the arithmetic sequence formula, the common difference makes the sequence linear, meaning the graph of the sequence terms is a straight line. Recognizing this pattern helps in verifying if a sequence is indeed arithmetic.

Nth Term Formula

Understanding the nth term formula in arithmetic sequences is essential for finding any specific term without listing all the previous ones.

The formula is expressed as:

• \( a_n = a_1 + (n-1)d \)

Where \( a_n \) represents the nth term, \( a_1 \) is the first term, "n" is the term number, and "d" is the common difference. Using this formula is like having a shortcut or a map to jump directly to any term's value in the sequence.

By understanding this formula, you can efficiently determine the position of terms in a sequence and cross-check if a number belongs to a given sequence. It simplifies the process of investigating sequences, especially when working with large numbers or extensive sequences.