Problem 32 - Algebra and Trigonometry

Question

find each indicated sum. $$ \sum_{i=1}^{5} i^{3} $$

Step-by-Step Solution

Answer

The sum of the cubes of the first 5 positive integers is 225.

Step 1: Identify the sequence

The sequence here is \(i^{3}\), for \(i\) ranging from 1 to 5. This means each term to be added will be a number cubed.

Step 2: Evaluate each term

Now, calculate each term in the sum: \(1^{3} = 1; 2^{3} = 8; 3^{3} = 27; 4^{3} = 64; 5^{3} = 125\)

Step 3: Add up the terms

Add all the evaluated terms together: \(1 + 8 + 27 + 64 + 125 = 225\)

Key Concepts

Sequences and Series

When we talk about sequences, we're referring to a set of numbers arranged in a particular order that follow a specific pattern. Each number in this list is called a term. For example, the sequence of cube numbers would include 1, 8, 27, and so on, where each term represents a number raised to the third power.

Series, on the other hand, is the summation of the terms of a sequence. If you're given a sequence, and you add all the terms together, you get a series. The summation notation \( \sum \) is a shorthand way to represent the addition of a sequence of numbers. In our exercise, the notation \( \sum_{i=1}^{5} i^{3} \) is asking for the sum of the first five terms of the sequence of cube numbers, which we find by calculating each term \( i^{3} \) where \( i \) ranges from 1 to 5 and then adding those terms together.

Exponents and Powers

Exponents and powers are a way of expressing repeated multiplication. An exponent, written as a small number above and to the right of a base number, tells you how many times to multiply the base by itself. For instance, \( 2^3 \) means \( 2 \) multiplied by itself three times, which equals 8.

On our current problem, the sequence is based on cube numbers, which means each term is a number raised to the power of 3. Here's how it looks for the sequence from 1 to 5: \( 1^3 = 1 \) ; \( 2^3 = 8 \) ; \( 3^3 = 27 \) ; \( 4^3 = 64 \) ; \( 5^3 = 125 \) . Understanding how to work with exponents is crucial for solving problems involving powers, such as the one in our exercise.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence where each term after the first is obtained by adding a constant, called the common difference, to the previous term. However, not all series are arithmetic. For instance, in the exercise, the series is not arithmetic because the difference between the terms is not constant.

Nonetheless, the concept of series still applies. To solve for the sum of any series, we need to evaluate and then sum up each term in the sequence. In arithmetic series, there's a handy formula we can use to find the sum quickly, but in the case of the series of cubes, we need to calculate each power individually and add them together, just as done in the exercise: \(1 + 8 + 27 + 64 + 125 = 225\).

Understanding the Distinction

Recognizing whether a series is arithmetic or not is important, as it determines the approach and formulas you can apply to find the sum.