Problem 63 - Algebra and Trigonometry

Question

\(61-68\) Write the sum using sigma notation. $$ 1^{2}+2^{2}+3^{2}+\cdots+10^{2} $$

Step-by-Step Solution

Answer

The sum is \(\sum_{i=1}^{10} i^2\).

Step 1: Identify the Pattern

The given sequence is the sum of squares of consecutive integers starting from 1 up to 10: \(1^2, 2^2, 3^2, \ldots, 10^2\). This pattern suggests the use of a summation of squares formula.

Step 2: Determine the Index of Summation

In sigma notation, the index of summation (often denoted by \(i\) or \(n\)) is used as a placeholder for each integer in the sequence. Here, we'll use \(i\) and it will range from 1 to 10.

Step 3: Write the Sigma Notation

In sigma notation, the sequence \(1^2 + 2^2 + 3^2 + \cdots + 10^2\) is expressed as the sum:\[\sum_{i=1}^{10} i^2\] This notation indicates that you sum the squares of \(i\) as it varies from 1 to 10.

Key Concepts

Summation of Squares

The summation of squares involves adding up the squares of sequential numbers. In our exercise, the numbers from 1 to 10 are squared and summed together. This means we look at numbers like 1 squared, 2 squared, all the way up to 10 squared. When we calculate these squares, we find numbers like 1, 4, 9, and so on.Using sigma notation, we write this sum as:

• The symbol \( \sum \) (sigma) tells us to sum things up.
• The expression \( i^2 \) means we are squaring each term \( i \).
• Our starting point \( i = 1 \) up to the end point \( i = 10 \) defines the sequence of numbers to square and sum.

Summation of squares is a common concept in mathematics, important for exploring patterns and understanding series in more advanced math topics.

Sequence

A sequence is a set of numbers arranged in a specific order. In the context of our exercise, it consists of squares of consecutive integers: \( 1^2, 2^2, 3^2, \ldots, 10^2 \). This sequence represents the ordered steps of squaring each successive number starting from 1.Mathematically, when we consider a sequence:

• Each number, or term, follows a particular pattern, here it is squaring the integer.
• The order of the terms matters, as it aligns with the concept of accumulation or addition sequentially.
• This pattern is visually summarized using sigma notation, allowing a compact and efficient representation of potentially lengthy sums.

Understanding sequences is crucial as they form the foundation for more complex mathematical concepts such as functions and series.

Index of Summation

The index of summation is a vital part of sigma notation. It determines how the sum progresses across specific values. In our example, this index is represented by the letter \( i \).Here's how it works:

• The index \( i \) starts from 1, indicating the first term of our sequence, \( 1^2 \).
• It moves up sequentially to the ending number, 10 in this case, represented by \( 10^2 \).
• The increment of the index informs how each successive term is included in the summation.

Using an index of summation makes mathematical expressions more elegant and easier to manage, especially when dealing with large sets of data or extensive computations.

Sum of Integers

The sum of integers is a foundational concept that helps us understand more complex summation processes. In this exercise, we deal with a sum involving squared integers which complicates the basic concept slightly. Here's what we do:

• Each integer from the sequence gets squared, changing the growth rate of the sum compared to simple addition.
• The individual results of these squares are then added together serially, providing a grand total.
• This total gives insight into the cumulative effect of sequentially squaring numbers, a useful exercise in finding patterns and relationships in mathematical series.

Ultimately, mastering the sum of integers is pivotal, since it's applied across many branches of mathematics and related real-world applications.