Problem 61 - Algebra and Trigonometry

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 Sequence

Look at the sequence presented: \(1^{2}+2^{2}+3^{2}+ ext{...}+10^{2}\). This is a sequence of squares of the first 10 positive integers.

Step 2: Determine the General Term

Each term in the sequence can be represented as \(n^2\), where \(n\) is the index of summation. So, the general term is \(n^2\).

Step 3: Establish the Limits of Summation

Identify the starting and ending terms. The sequence starts at \(n = 1\) and ends at \(n = 10\). Thus, the lower limit is 1, and the upper limit is 10.

Step 4: Write the Sigma Notation

Using the information from the previous steps, represent the sum \(1^{2}+2^{2}+3^{2}+\ldots+10^{2}\) in sigma notation. It is written as: \[ \sum_{n=1}^{10} n^2 \].

Key Concepts

sequence of squares

Understanding the sequence of squares is fundamental to working with sums and sigma notation. A sequence is a set of numbers arranged in a specific order. The sequence of squares refers to numbers that are each multiplied by themselves, such as 1 squared, 2 squared, and so on. In mathematical terms, this means each number in the sequence is raised to the power of 2, or squared.

• First Term: The sequence of squares starts with the smallest square, which is 1, the square of 1 ( 1^2 = 1).
• Next Terms: Each subsequent term is also a square: 4 ( 2^2), 9 ( 3^2), 16 ( 4^2), continuing in that pattern.
• General Pattern: For each integer n, the n-th term is given by n squared ( n^2).

These sequences appear frequently in various mathematical contexts, such as in the study of quadratic functions and number theory. Recognizing the pattern is essential for operations like summation using sigma notation.

sum of squares

The sum of squares involves adding together the results of a sequence of squared integers. Summing these values tells us the total of all individual squares up to a specific number. This concept is widely used in statistics, physics, and many other fields.

• Expanding the Formula: In our example, the sum is the result of adding numbers like 1 ( 1^2), 4 ( 2^2), 9 ( 3^2), up to the last term, 100 ( 10^2).
• Applications: Beyond pure mathematics, sum of squares calculations are crucial in statistical methods like variance and standard deviation, where differences between observed values and the mean are squared and summed.
• Benefits of Summing: This method helps simplify expressions and determine overall tendencies or total magnitude more effectively.

Summing squares can often provide a straightforward way to reach a large number of results. It captures the essence of how much overall square values contribute to the total.

limits of summation

Determining the limits of summation is crucial when representing series using sigma notation. These limits define the boundaries within which the summation occurs, telling us where to start and stop. Understanding these limits is essential when dealing with sequences and their sums.

• Lower Limit: This is the starting point of your summation. In our example, the lower limit is set at 1 (the first square in the series)
.
• Upper Limit: This is where your summation ends. The upper limit in our sequence is 10, concluding with 10^2.
• Importance: Correctly setting these limits ensures that the sequence is covered entirely without missing or overcounting any term.

Knowing the correct limits helps in writing the sigma notation accurately. Setting these boundaries lets you represent complex sequences in a condensed, manageable way. Limits of summation are thus pivotal in grasping the breadth of a series efficiently.