Problem 2 - Precalculus: Mathematics for Calculus

Question

The \(n\) th partial sum of a sequence is the sum of the first __________ terms of the sequence. So for the sequence \(a_{n}=n^{2}\) the fourth partial sum is \(S_{4}=\) _____ \(+\) ______ \(+\) ______ \(+\) _____ \(=\) _____ .

Step-by-Step Solution

Answer

The fourth partial sum is 30.

Step 1: Identify the Sequence

The sequence provided is represented as \(a_n = n^2\). This means that each term of the sequence is the square of its position number.

Step 2: Define the Partial Sum

The \(n\)th partial sum of a sequence is the sum of the first \(n\) terms. To find the fourth partial sum \(S_4\), we need to sum the first four terms of the sequence \(a_n = n^2\).

Step 3: Calculate Each Term

Calculate the first four terms: \(a_1 = 1^2 = 1\), \(a_2 = 2^2 = 4\), \(a_3 = 3^2 = 9\), \(a_4 = 4^2 = 16\).

Step 4: Sum the Terms

Add the first four terms to find \(S_4\): \(S_4 = 1 + 4 + 9 + 16\).

Step 5: Compute the Total

Calculate the total: \(S_4 = 1 + 4 + 9 + 16 = 30\).

Key Concepts

Understanding Sequences

A sequence is an ordered list of numbers derived from a specific rule or pattern. The power of a sequence lies in its predictability and the ease with which future terms can be defined once this rule is known. For the sequence given in the exercise, each term is represented by the formula \(a_n = n^2\). This means that for any position \(n\) in the sequence, the corresponding term is the square of \(n\).

• The first term, when \(n = 1\), is \(a_1 = 1^2 = 1\).
• The second term, when \(n = 2\), is \(a_2 = 2^2 = 4\).
• The third term, when \(n = 3\), is \(a_3 = 3^2 = 9\).
• The fourth term, when \(n = 4\), is \(a_4 = 4^2 = 16\).

Sequences are fundamental in mathematics, as they form the groundwork for more complex concepts such as series and partial sums.

Exploring Squared Terms

Squared terms are a specific type of mathematical expression where a number is multiplied by itself. In the context of the sequence \(a_n = n^2\), each term in the sequence is a squared term. Understanding squared terms is crucial because they appear frequently not only in algebra but also in geometry and various scientific calculations.

• Calculate squared terms by multiplying a number by itself: \(n^2 = n \times n\).
• Recognize patterns, such as how the difference between consecutive squared terms \((n+1)^2 - n^2\) forms arithmetic series, i.e., consecutive odd numbers \(2n + 1\).

This sequencing with squared numbers creates a distinctive quadratic progression, showcasing the symmetry and incremental growth characteristic of squares.

Sum of a Series Using Partial Sums

The sum of a series, specifically a partial sum, refers to adding a finite number of terms from a sequence. In the exercise, the series derived from the sequence \(a_n = n^2\) is summed up to find the fourth partial sum, denoted as \(S_4\). Calculating a partial sum involves the following steps:

• Identify the terms to be included: Here, the terms are \(1, 4, 9, 16\).
• Add these terms together: \(S_4 = 1 + 4 + 9 + 16\).
• The result gives the partial sum: \(S_4 = 30\).

Understanding partial sums is crucial as it allows learners to approach infinite series by examining their finite approximations, serving as a foundational concept in calculus and numerical analysis.