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Chapter 11: Problem 17

Find the sum of each series. \(\sum_{i=0}^{5} i(i-1)(i-2)\)

Short Answer

Expert verified
The sum is 90.

Step by step solution

01

- Understand the Series

The series given is \(\sum_{i=0}^{5} i(i-1)(i-2)\). This means we need to find the sum of the expression \(i(i-1)(i-2)\) starting from \(i = 0\) and ending at \(i = 5\).
02

- Compute Each Term

Evaluate each term separately: \For \(i = 0\): \(0(0-1)(0-2) = 0\) \For \(i = 1\): \(1(1-1)(1-2) = 0\) \For \(i = 2\): \(2(2-1)(2-2) = 0\) \For \(i = 3\): \(3(3-1)(3-2) = 6\) \For \(i = 4\): \(4(4-1)(4-2) = 24\) \For \(i = 5\): \(5(5-1)(5-2) = 60\)
03

- Sum All the Terms

Add the results of each term: \(0 + 0 + 0 + 6 + 24 + 60 = 90\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mathematical series
A mathematical series is the sum of the terms of a sequence of numbers. The sequence is an ordered list of numbers, and when we sum these numbers, we get a series. Mathematical series are used in many areas of mathematics and science, as they help to represent data and solve problems more efficiently.
One common example of a mathematical series is the arithmetic series, where the difference between consecutive terms is constant. Another example is the geometric series, where each term is a constant multiple of the previous term.
In the given exercise, the series involves polynomial expressions, where each term is calculated using a formula which itself involves a polynomial.
sum calculation
Sum calculation is the process of adding together multiple numbers or expressions to get a single result. This is a basic but essential operation in mathematics.
In the exercise, we are calculating the sum of a series \(\textstyle \sum_{i=0}^{5} i(i-1)(i-2)\). This notation indicates that we need to evaluate the expression \(i(i-1)(i-2)\) for each value of \(i\) from 0 to 5, and then add all these values together:
    - Start by substituting \(i\) into the expression one at a time.
    - Evaluate the expression for each \(i\).
    - Finally, add all the evaluated results to get the sum.
This simple method works effectively for small series but can become more complex for larger series or different expressions.
polynomial expressions
A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, the expression \(x^2 + 2x + 1\) is a polynomial.
In the context of the given exercise, the expression \(i(i-1)(i-2)\) is a polynomial in the variable \(i\). It has three terms that involve the variable:
    - \(i\) which is a linear term.
    - \(i-1\) which is another linear term.
    - \(i-2\) which is one more linear term.
When these terms are multiplied together, they form a product which is also a polynomial, specifically a cubic polynomial as the highest degree term will be \(i^3\). By evaluating this polynomial for different values of \(i\) and summing the results, we can compute the sum of the series as demonstrated in the solution.

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