Problem 65 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5}\left(4 i^{2}-2 i+6\right)$$

Step-by-Step Solution

Answer

The sum of the series is 220.

Step 1: Understanding the Series

The given series is \( \sum_{i=1}^{5}(4i^2 - 2i + 6) \). This means we need to find the sum of the expression \( 4i^2 - 2i + 6 \) as \( i \) goes from 1 to 5.

Step 2: Substitute and Simplify Each Term

We substitute each integer from 1 to 5 into the expression \( 4i^2 - 2i + 6 \) to calculate the individual terms:- For \( i = 1 \): \( 4(1)^2 - 2(1) + 6 = 4 - 2 + 6 = 8 \)- For \( i = 2 \): \( 4(2)^2 - 2(2) + 6 = 16 - 4 + 6 = 18 \)- For \( i = 3 \): \( 4(3)^2 - 2(3) + 6 = 36 - 6 + 6 = 36 \)- For \( i = 4 \): \( 4(4)^2 - 2(4) + 6 = 64 - 8 + 6 = 62 \)- For \( i = 5 \): \( 4(5)^2 - 2(5) + 6 = 100 - 10 + 6 = 96 \)

Step 3: Add the Terms Together

Sum the results of each substitution to find the total sum for the series:\( 8 + 18 + 36 + 62 + 96 = 220 \).

Key Concepts

Substitution Method

The substitution method is an essential process in solving arithmetic series. It involves replacing variables with specific numbers to find individual term values in the series. When you encounter a series like the one given, \( \sum_{i=1}^{5}(4i^2 - 2i + 6) \), substitution means you will systematically replace \( i \) with each integer from the defined limits—in this case, 1 through 5.

• This process helps in breaking down a complicated expression into manageable pieces.
• Each substitution results in a distinct number, representing a term in the series.
• By finding each term separately, the series becomes easier to understand and calculate.

Try substituting values thoughtfully to avoid arithmetic mistakes. Once each term is calculated, it becomes straightforward to add them together to find the series sum.

Summation Notation

Summation notation is a concise way of expressing the addition of a sequence of numbers. The symbol \( \sum \) denotes the act of summing, and it's followed by an expression that includes a variable, such as \( i \). The series \( \sum_{i=1}^{5}(4i^2 - 2i + 6) \) demonstrates the key parts:

• The variable under the summation symbol, \( i \), signifies the index of summation.
• The numbers 1 and 5 indicate the starting and ending values for this index.
• The expression inside, \( 4i^2 - 2i + 6 \), is the formula for generating the terms.

Understanding summation notation helps you quickly identify how many terms need calculation and what pattern each term follows. It simplifies complex arrangements into readable forms.

Algebraic Expressions

An algebraic expression is a vital element in arithmetic. It involves equations like \( 4i^2 - 2i + 6 \) in the problem. These expressions consist of variables, constants, and arithmetic operations. To break down what happens inside:

• \( 4i^2 \) is a term where \( i \) is squared, then multiplied by 4, making it quadratic.
• \( -2i \) brings the linear component, meaning \( i \) is reduced by 2.
• \( +6 \) is the constant, influencing each term equally regardless of \( i \).

When solving the series, manipulating this expression for each \( i \) allows it to reveal its respective term. Grasping how each element interacts within an algebraic expression aids in understanding and predicting results.