Problem 67 - A Graphical Approach to Precalculus with Limits

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: Understand the Series Formula

The expression to sum is \(4i^2 - 2i + 6\). We are tasked with finding the total sum from \(i = 1\) to \(i = 5\).

Step 2: Substitute and Evaluate for Each Term

Substitute each value of \(i\) from 1 to 5 into the expression \(4i^2 - 2i + 6\). Calculate each term's result.

Step 2.1: Substitute i = 1

For \(i = 1\), substitute into the expression: \[4(1)^2 - 2(1) + 6 = 4 - 2 + 6 = 8\]

Step 2.2: Substitute i = 2

For \(i = 2\), substitute into the expression: \[4(2)^2 - 2(2) + 6 = 16 - 4 + 6 = 18\]

Step 2.3: Substitute i = 3

For \(i = 3\), substitute into the expression: \[4(3)^2 - 2(3) + 6 = 36 - 6 + 6 = 36\]

Step 2.4: Substitute i = 4

For \(i = 4\), substitute into the expression: \[4(4)^2 - 2(4) + 6 = 64 - 8 + 6 = 62\]

Step 2.5: Substitute i = 5

For \(i = 5\), substitute into the expression: \[4(5)^2 - 2(5) + 6 = 100 - 10 + 6 = 96\]

Step 3: Compute the Sum of the Series

Add the computed values from each term: \[8 + 18 + 36 + 62 + 96 = 220\]

Step 4: Verify Your Result

Double-check each step to ensure the calculations are correct and that each sum term was accurately evaluated.

Key Concepts

Arithmetic series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is consistent. This is known as the common difference. In arithmetic series, you typically find the sum by identifying the first term, the last term, and the number of terms in the series. Then, it's a matter of applying the formula:

• The sum of an arithmetic series is given by \( S_n = \frac{n}{2} (a + l) \), where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

Although the given exercise doesn’t present a typical arithmetic series since the terms are governed by a polynomial expression, understanding arithmetic progressions helps in visualizing how different terms accumulate to form a series.An understanding of arithmetic series is fundamental when faced with more complex summations, as it teaches the basics of adding sequential terms with uniform differences.

Polynomial expressions

Polynomial expressions are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and positive integer exponents of variables. A typical polynomial looks like this:

• \( a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 \)

In the given exercise, the polynomial expression in play is \( 4i^2 - 2i + 6 \). Polynomials can be evaluated for various values of their variable by substituting the specific values into the expression and computing the result.Polynomials are crucial in this context as they define the relationship between terms within a series and play a significant role when determining the behavior of the series as more terms are added.

Evaluation of a series

Evaluation of a series involves the process of calculating and compiling the sum of a sequence of expressions. Here's what you generally do:

• Identify the expression you'll work with. In our case, it's \( 4i^2 - 2i + 6 \).
• Substitute consecutive integer values beginning from the lower limit up to the upper limit described by the summation notation.
• Calculate each individual term.

For this exercise, you compute separately for \( i \) from 1 to 5, resulting in: \[ 8, 18, 36, 62, \text{and} \, 96 \].After calculating and aligning each term, you simply sum these assessed values to get the series' total sum.Evaluating each part carefully ensures that any mix-ups or miscalculations can be intercepted promptly, leading to the exact sum of the series as \( 220 \).

Summation notation

Summation notation is a concise way to express sums with several terms. It uses the symbol \( \Sigma \) (capital Greek letter Sigma) to sum terms according to specified indices. The expression below \( \Sigma \) indicates the elements to be summed, and often the variable indicates ranging indices:

• The general structure: \( \sum_{i=a}^{b}x_i \), where \( i \) starts at \( a \) and ends at \( b \).

In this particular exercise, it sums the expression \( 4i^2 - 2i + 6 \) as \( i \) takes on integers from 1 to 5.By using this notation, calculations can be neatly organized, giving a clear acknowledgment of when and how each part of the series is calculated.Summation notation provides mathematicians and students a powerful tool to efficiently convey repetitive additions in a visually efficient format.