Problem 68 - A Graphical Approach to College Algebra

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

The given series is \( \sum_{i=1}^{6}(i^2 + 2i^3) \). This means we need to compute the expression \( i^2 + 2i^3 \) for each integer value of \( i \) from 1 to 6, and then add up all these computed values.

Step 2: Calculate Each Term

Calculate the expression \( i^2 + 2i^3 \) for each value of \( i \):- For \( i = 1 \), \( 1^2 + 2(1)^3 = 1 + 2 = 3 \).- For \( i = 2 \), \( 2^2 + 2(2)^3 = 4 + 16 = 20 \).- For \( i = 3 \), \( 3^2 + 2(3)^3 = 9 + 54 = 63 \).- For \( i = 4 \), \( 4^2 + 2(4)^3 = 16 + 128 = 144 \).- For \( i = 5 \), \( 5^2 + 2(5)^3 = 25 + 250 = 275 \).- For \( i = 6 \), \( 6^2 + 2(6)^3 = 36 + 432 = 468 \).

Step 3: Sum All Calculated Terms

Add all the calculated terms together:\[ 3 + 20 + 63 + 144 + 275 + 468 \]

Step 4: Compute the Total Sum

Perform the actual addition:\[ 3 + 20 + 63 + 144 + 275 + 468 = 973 \]

Step 5: Write the Final Answer

Thus, the sum of the series \( \sum_{i=1}^{6}(i^2 + 2i^3) \) is \( 973 \).

Key Concepts

Understanding Sigma Notation

Sigma notation is a powerful and compact way to represent the sum of a sequence of terms. It uses the Greek letter Sigma (\( \Sigma \)) to denote summation. In our problem, we encountered sigma notation in the format \( \sum_{i=1}^{6}(i^2 + 2i^3) \), which means you need to calculate the sum of values generated by plugging integers from 1 to 6 into the expression \( i^2 + 2i^3 \).

This notation involves three parts:

• The variable of summation (often \( i \)), which represents the terms to vary.
• The range of the summation (1 to 6 in our case) indicates the integer values the variable will take.
• The expression \( i^2 + 2i^3 \) that will be evaluated for each value of \( i \).

Using sigma notation helps simplify the representation of complex sums by avoiding the need to write out each term individually. It is essential to understand how to set up and interpret this notation for clarity in mathematics.

Exploring Polynomial Expressions

A polynomial expression like \( i^2 + 2i^3 \) is a combination of terms where each term includes a variable raised to a whole number exponent. In this case, the polynomial includes squared \((i^2)\) and cubed \((i^3)\) variables.

Here’s a breakdown:

• \( i^2 \) has a degree of 2, making it a quadratic term.
• \( 2i^3 \) has a degree of 3, categorizing it as a cubic term.
• The coefficients, 1 for \( i^2 \) and 2 for \( i^3 \), multiply their respective terms.

These expressions are fundamental in algebra and calculus and can model various physical phenomena due to their flexibility in form. Understanding polynomial expressions helps in solving, describing, and expanding many mathematical problems.

Deciphering Mathematical Series

A mathematical series is essentially the sum of the terms of a sequence. In a series, each term is defined systematically, often by a specific formula.

Consider the series described by \( \sum_{i=1}^{6}(i^2 + 2i^3) \):

• This series has 6 terms, ranging from \( i = 1 \) to \( i = 6 \).
• Each term is defined by the formula \( i^2 + 2i^3 \), which you compute by substituting each integer from the range into the formula.

The beauty of a mathematical series is in its predictability and structure. Recognizing a series and being able to compute its sum by resolving each term individually is a key in many mathematical applications.

Mastering Summation Techniques

Summation techniques revolve around finding the sum of sequences efficiently using formulated methods or properties. In this exercise, we solved the sum \( \sum_{i=1}^{6}(i^2 + 2i^3) \) by calculating each term and then adding them together.

Here are some key summation techniques:

• **Arithmetic Series Formula:** Useful for sequences where each term increases by a constant.
• **Geometric Series Formula:** Efficient for sequences where each term is multiplied by a constant to get the next.
• **Direct Summation:** Like in our problem, individually calculate and sum each term.

Understanding when and how to apply these techniques can save time and simplify complex calculations in algebra and calculus contexts. Direct summation, as demonstrated in this exercise, is straightforward and ensures accuracy for small sequence ranges.