Problem 60 - Algebra and Trigonometry

Question

Express each sum using summation notation. \(1^{3}+2^{3}+3^{3}+\cdots+8^{3}\)

Step-by-Step Solution

Answer

\(\sum_{i=1}^{8} i^3\)

Step 1: Identify the Pattern

Observe the given sequence: \[1^3 + 2^3 + 3^3 + \ldots + 8^3\]. Each term in the sequence is the cube of an integer.

Step 2: Establish the General Term

Recognize that each term in the sequence can be expressed as \(i^3\) where \(i\) is the integer that ranges from 1 to 8.

Step 3: Use Summation Notation

Express the sequence using the summation notation: \[\sum_{i=1}^{8} i^3\]. This notation indicates that we are summing the cubes of the integers from 1 to 8.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In this exercise, we see algebra at play through the use of notation and the manipulation of sequences.
Understanding algebra involves recognizing patterns, establishing general terms, and using symbols to represent mathematical operations.
Here, the sequence of cubes is represented as a series of algebraic expressions, which simplifies the process of understanding and summation.

Sequences and Series

Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers and their summation, respectively.
A sequence is simply a set of numbers arranged in a specific order, while a series is the sum of the elements in a sequence.
In the given exercise, the sequence is 1^3, 2^3, 3^3, ..., 8^3. Each term follows a consistent pattern—it is the cube of consecutive integers from 1 to 8.
This ordered arrangement makes it easier to recognize and express the general term, which in this case, is the cube of any integer within the sequence.

Summation

Summation, often represented by the Greek letter \(\forall\), is the process of adding up a sequence of numbers.
The exercise above illustrates this concept by summing up the cubes of integers from 1 to 8.
Summation notation \(\sum_{i=1}^{8} i^3 \) is used to succinctly represent this process.
It tells us to add the values of \(i^3\) for each integer value of \(i\) from 1 to 8, making it a powerful tool for simplifying complex addition.