Problem 63 - Precalculus Mathematics for Calculus

Question

Write the sum using sigma notation. $$1^{2}+2^{2}+3^{2}+\dots+10^{2}$$

Step-by-Step Solution

Answer

\(\sum_{n=1}^{10} n^2\)

Step 1: Identify the Pattern

The problem presents a sequence of squared numbers: \(1^2, 2^2, 3^2, \ldots, 10^2\). Notice that each term in the sequence is a square of an integer.

Step 2: Determine the General Term

The general term for the sequence can be expressed as \(n^2\), where \(n\) is each integer from 1 to 10.

Step 3: Set the Limits for Summation

The sequence starts at \(n=1\) and ends at \(n=10\). This means the summation will start at 1 and go up to 10.

Step 4: Write Using Sigma Notation

Using the general term \(n^2\) and the limits from 1 to 10, the sum can be written in sigma notation as: \[\sum_{n=1}^{10} n^2\]

Key Concepts

Sum of Squares

In mathematics, the **sum of squares** refers to the sum of the squares of different numbers. The exercise above provides an example where numbers from 1 to 10 are squared and then summed. Each term in the sequence, such as \(1^2, 2^2, 3^2, \ldots, 10^2\), forms a sum of squares.This concept is highly significant in various mathematical studies and applications, including statistics (e.g., variance calculations) and geometry. If you have several numbers \(x_1, x_2, \ldots, x_n\), the sum of squares is represented as:\[x_1^2 + x_2^2 + \cdots + x_n^2\]The importance of understanding the sum of squares is that it helps in analyzing numerical data. It's used in formulas for calculating standard deviation and other statistical measures which assess variation within data.

Sequences and Series

Sequences and series form a fundamental part of mathematical analysis. A **sequence** is essentially a list of numbers arranged in a specific order. For example, in the exercise, the sequence is \(1^2, 2^2, 3^2, \ldots, 10^2\). Each number is a term of the sequence and is defined by an associated formula.A **series**, on the other hand, is the sum of the terms of a sequence. Here, the series is formed by adding the sequence's terms: \(1^2 + 2^2 + 3^2 + \cdots + 10^2\).Understanding sequences and series is crucial because:

• They help in defining the behavior of functions and mathematical models over time.
• They are used to solve problems in calculus and algebra.
• Recognizing patterns in sequences allows predicting and generalizing mathematical phenomena.

These concepts not only aid in comprehending the given exercise but also open doors to learning advanced topics like differential equations and infinite series.

Mathematical Notation

**Mathematical notation** is a symbolic language used to express mathematical ideas with precision and clarity. In the exercise, we've seen sigma notation, represented by the Greek letter \(\Sigma\). This specific notation is employed to represent the sum of a sequence of terms.The sigma notation in the solution is written as:\[\sum_{n=1}^{10} n^2\]Here's how to understand this notation:

• The \(\Sigma\) indicates a summation.
• \(n=1\) is the starting index, meaning the summation starts from when \(n\) equals 1.
• \(10\) is the ending index, indicating that \(n\) goes up to 10.
• \(n^2\) is the general term of the sequence being summed.

This compact notation helps describe complex sums succinctly and is widely used in calculus, statistics, and many other fields of mathematics. Mastery of this notation allows students to understand and convey mathematical concepts more efficiently.