Problem 11 - College Algebra with Modeling and Visualization

Question

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=n^{2}+1 $$

Step-by-Step Solution

Answer

The sum of the first five terms, \( S_5 \), is 60.

Step 1: Understand the Series Formula

The given sequence formula is \( a_n = n^2 + 1 \). This formula will help us calculate each term of the sequence individually.

Step 2: Calculate the First Term, \(a_1\)

Substitute \( n = 1 \) into the formula to find the first term: \( a_1 = 1^2 + 1 = 2 \).

Step 3: Calculate the Second Term, \(a_2\)

Substitute \( n = 2 \) into the formula to find the second term: \( a_2 = 2^2 + 1 = 5 \).

Step 4: Calculate the Third Term, \(a_3\)

Substitute \( n = 3 \) into the formula to find the third term: \( a_3 = 3^2 + 1 = 10 \).

Step 5: Calculate the Fourth Term, \(a_4\)

Substitute \( n = 4 \) into the formula to find the fourth term: \( a_4 = 4^2 + 1 = 17 \).

Step 6: Calculate the Fifth Term, \(a_5\)

Substitute \( n = 5 \) into the formula to find the fifth term: \( a_5 = 5^2 + 1 = 26 \).

Step 7: Sum the First Five Terms

Now, sum the five terms \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 2 + 5 + 10 + 17 + 26 \).

Step 8: Simplify the Sum

Add the values: \( 2 + 5 + 10 + 17 + 26 = 60 \). Thus, \( S_5 = 60 \).

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that helps us find each term in a sequence. In our given exercise, the sequence formula is presented as \( a_n = n^2 + 1 \), where \( n \) is a positive integer representing the position of the term in the sequence. With \( a_n = n^2 + 1 \), for any position \( n \), we can compute the term by substituting \( n \) into this formula. This formula allows us to see patterns, structures, and ultimately define the terms of the sequence step-by-step.Let's break this down:

• When \( n = 1 \), \( a_1 = 1^2 + 1 = 2 \)
• When \( n = 2 \), \( a_2 = 2^2 + 1 = 5 \)
• When \( n = 3 \), \( a_3 = 3^2 + 1 = 10 \)
• ...and so forth for any term \( a_n \).

Understanding the sequence formula is the first step in tackling problems related to sequences, as it provides a concrete method to calculate each term's value individually.

Summation

Summation is the process of adding a sequence of numbers, which often results in finding the Partial Sum or Total Sum of terms. In the context of our exercise, we need to find the sum of the first five terms given by the sequence formula \( a_n = n^2 + 1 \).This means calculating the terms individually first:- \( a_1 = 2 \)- \( a_2 = 5 \)- \( a_3 = 10 \)- \( a_4 = 17 \)- \( a_5 = 26 \)Next, we sum these terms:\[ S_5 = 2 + 5 + 10 + 17 + 26 \]When summed together, the terms give a total of \( S_5 = 60 \). This process of adding terms together to get a total is key in series and sequences, particularly in finding how large a function behaves as it's summed over larger values.

Arithmetic Operations

Arithmetic operations involve basic calculations including addition, subtraction, multiplication, and division. These operations are foundational skills in mathematics and are heavily applied when working with sequences and series.Let's see how they feature in our exercise:- **Addition**: Used to sum the sequence terms to find the total or partial sum, as shown: \( S_5 = 2 + 5 + 10 + 17 + 26 \).- **Multiplication**: When substituting \( n \) into the formula \( n^2 \), multiplication is used to find terms like \( n^2 \), e.g., \( 5^2 \).Each operation is straightforward:

• Adding numbers gives a total.
• Multiplying a number by itself computes squares (\( n^2 \) in this exercise).

These operations serve as simple yet critical tools needed to compute and handle numerical values within sequences and series effectively. Without mastering these operations, working through sequences would be much more challenging.