Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=1}^{6}\left(i^{2}+2 i^{3}\right)$$

Step-by-Step Solution

Answer

The sum of the series is 973.

Step 1: Understand the Series Structure

The series given is \( \sum_{i=1}^{6} (i^2 + 2i^3) \). This means that for each integer \( i \) from 1 to 6, you calculate \( i^2 + 2i^3 \) and then sum all these values.

Step 2: Calculate Each Term

Calculate each term in the series by substituting values of \( i \) from 1 to 6:- For \( i = 1 \), term is \( 1^2 + 2 \times 1^3 = 1 + 2 = 3 \)- For \( i = 2 \), term is \( 2^2 + 2 \times 2^3 = 4 + 16 = 20 \)- For \( i = 3 \), term is \( 3^2 + 2 \times 3^3 = 9 + 54 = 63 \)- For \( i = 4 \), term is \( 4^2 + 2 \times 4^3 = 16 + 128 = 144 \)- For \( i = 5 \), term is \( 5^2 + 2 \times 5^3 = 25 + 250 = 275 \)- For \( i = 6 \), term is \( 6^2 + 2 \times 6^3 = 36 + 432 = 468 \)

Step 3: Sum All Terms

Add all the calculated terms together: - Sum = 3 + 20 + 63 + 144 + 275 + 468 = 973.

Key Concepts

Summation Notation

Summation notation is a compact way to represent the addition of a sequence of numbers. It is typically written using the Greek letter sigma (\( \Sigma \)). In this context, the expression \( \sum_{i=1}^{6} (i^2 + 2i^3) \) means you start by setting \( i = 1 \) and go up to \( i = 6 \).
For each value of \( i \), you compute \( i^2 + 2i^3 \) and sum these results. Using summation notation helps keep complex calculations organized and easily readable, which simplifies working with series calculations.

• \(i = 1 \to i^2 + 2i^3 = 1^2 + 2 \times 1^3 = 3\)
• \(i = 2 \to i^2 + 2i^3 = 2^2 + 2 \times 2^3 = 20\)
• Continue similarly for each \(i\) up to 6.

Remember, the numbers represent terms that form the series, offering a method to systematically add them.

Polynomial Expressions

A polynomial expression is an algebraic expression that consists of variables and coefficients, structured in terms of powers and sums. In the problem given, the expression inside the summation notation \((i^2 + 2i^3)\) is a polynomial in terms of \(i\).
Polynomials are classified based on their degree, which is the highest power of the variable in the expression. For instance, in \(i^2 + 2i^3\):

• The term \(i^2\) is a polynomial of degree 2.
• The term \(2i^3\) is of degree 3, making it the dominant term of the polynomial expression.

These polynomial expressions, especially in series, can represent more complex relationships and increase the variety of sequences that one can sum, making them versatile for many mathematical applications.

Series Calculation

Series calculation involves summing a sequence of numbers or terms that follow a specific pattern or formula. The series in this exercise is expressed as \( \sum_{i=1}^{6} (i^2 + 2i^3) \), where each term is calculated based on the specified formula before summing them together.
The process breaks down into manageable steps:

• Substitute values: Insert each integer from the defined range into the polynomial expression \((i^2 + 2i^3)\).
• Calculate each term: Compute values for each individual \(i\): \(3, 20, 63, 144, 275, 468\).
• Add the terms: Summing these results: \(3 + 20 + 63 + 144 + 275 + 468\) gives the total \(973\).

Series calculations can be found in statistics, calculus, and financial mathematics, showing how crucial understanding this process is for broader applications.