Problem 43 - Contemporary Precalculus

Question

Find the sum. $$\sum_{i=1}^{4}\left(i^{2}+1\right)$$

Step-by-Step Solution

Answer

Question: Calculate the sum of the series $\sum_{i=1}^{4}\left(i^{2}+1\right)$. Answer: The sum of the series $\sum_{i=1}^{4}\left(i^{2}+1\right)$ is 34.

Step 1: Identify the value range of "i"

The variable "i" will take values from 1 to 4 as given by the summation notation. So we will evaluate the expression inside the summation for i=1, i=2, i=3, and i=4, and add up all the results.

Step 2: Substitute i=1 into the expression

Plugging in i=1 into the expression, we get: $$(1^2 + 1) = (1 + 1) = 2$$

Step 3: Substitute i=2 into the expression

Now, let's plug in i=2 into the expression: $$(2^2 + 1) = (4 + 1) = 5$$

Step 4: Substitute i=3 into the expression

Next, substitute i=3 into the expression: $$(3^2 + 1) = (9 + 1) = 10$$

Step 5: Substitute i=4 into the expression

Finally, let's plug in i=4 into the expression: $$(4^2 + 1) = (16 + 1) = 17$$

Step 6: Calculate the sum of the results

Now let's add up all the results obtained: $$\sum_{i=1}^{4}\left(i^{2}+1\right) = 2+5+10+17 = 34$$

Step 7: State the final answer

Therefore, the sum of the series is $$\sum_{i=1}^{4}\left(i^{2}+1\right)= 34$$.

Key Concepts

Series

In mathematics, a series is the sum of the terms of a sequence. The series is denoted by the summation symbol and signifies the sum of the sequence elements from the first term to the last.
For example, when you see an expression like \( \sum_{i=1}^{n}a_i \), it means you are adding all the terms from \( a_1 \) to \( a_n \).
This concept is crucial as it allows one to work with a large set of data points efficiently.

• A sequence is just a list of numbers.
• When we add these numbers together, it becomes a series.
• Each element in the series corresponds to a specific position in the sequence.

The exercise you reviewed is an example of a mathematical series where each term consists of \( i^2 + 1 \). Here, each term is specific to its place in the sequence and impacts the total sum when added continuously from \( i=1 \) to \( i=4 \). Understanding series is important as it is used in calculus, physics, and finance to solve complex problems involving sums of infinite or finite terms.

Sigma Notation

Sigma notation, represented by the Greek letter \( \Sigma \), is a convenient way to express the sum of a series of terms. It's one of the most powerful notations in mathematics due to its compactness and clarity. In sigma notation, you specify the sequence terms you want to add, the starting and ending indices, and the expression for the terms in the sequence.
The general form is \( \Sigma_{i=a}^{b}f(i) \), where:

• \( i \) is the index of summation.
• \( a \) and \( b \) are the limits that indicate the range of the index.
• \( f(i) \) is the expression that depends on \( i \).

This notation makes it easy to see the structure and components of the series. It also facilitates understanding and computation, especially when dealing with complex expressions over extensive ranges. In the context of the exercise, \( \Sigma_{i=1}^{4}(i^2 + 1) \) clearly shows how the sum is to be evaluated from \( i=1 \) to \( i=4 \). This provides a straightforward way to organize and compute each evaluation step by step.

Arithmetic Operations

Arithmetic operations involve basic mathematical processes such as addition, subtraction, multiplication, and division. These operations are the foundation of tougher mathematical concepts like series and summation. Understanding and applying basic arithmetic helps in solving summation problems effectively. In the context of the problem given, arithmetic operations facilitate the calculation of terms in the series.
For example, to solve the series \( \Sigma_{i=1}^{4}\,(i^2 + 1) \), you perform the following operations:

• Square each \( i \) value (\( i^2 \)).
• Add 1 to each squared value from \( i = 1 \) to \( i = 4 \).
• Add up all these values from the computations.

By executing arithmetic operations sequentially as shown in the step-by-step solution, you systematically reach the total sum of 34. These operations are crucial for computing the terms and the final resultant sum in a series.