Problem 8 - Calculus: Early Transcendental Functions

Question

Write out all terms and compute the sums. $$\sum_{i=6}^{8}\left(i^{2}+2\right)$$

Step-by-Step Solution

Answer

The sum of the sequence is 155

Step 1: Calculate individual terms

The first step is to calculate each term in the sequence: \(6^{2}+2, 7^{2}+2, 8^{2}+2\), which becomes \(38, 51, 66\) respectively

Step 2: Calculate the sum

Next is to sum up all these terms. A sum of \(38 + 51 + 66\) which gives us a total of \(155\) as result.

Key Concepts

Sequence and Series

The concepts of sequences and series are fundamental in calculus and mathematics in general. A **sequence** is an ordered list of numbers, often generated by a specific rule or formula. These numbers are called terms.
In the given exercise, the terms of the sequence are generated by the formula \(i^2 + 2\), where \(i\) takes on consecutive integer values from 6 to 8:

• The first term is \(6^2 + 2 = 38\).
• The second term is \(7^2 + 2 = 51\).
• The third term is \(8^2 + 2 = 66\).

When we refer to a **series**, we are talking about the sum of the terms of a sequence. In this exercise, the series is the sum of the three terms: \(38 + 51 + 66 = 155\).
Sequences and series are used in calculus and other mathematical fields to analyze and compute various functions and mathematical models.

Step-by-Step Solutions

Step-by-step solutions are vital, especially in mathematics, to allow students to follow the logical progression of a problem. Breaking down a problem into smaller, manageable steps helps to clarify complex calculations.
Let's revisit the solution in the exercise:

• **Step 1:** Begin by determining each term; in our exercise, we calculate each term using the formula \(i^2 + 2\). We then plug in the values of \(i\) from 6 to 8: - When \(i = 6\): Calculate \(6^2 + 2 = 38\).
  - When \(i = 7\): Calculate \(7^2 + 2 = 51\).
  - When \(i = 8\): Calculate \(8^2 + 2 = 66\).
  
• **Step 2:** After determining each individual term, the final step is to compute the total sum: \(38 + 51 + 66 = 155\).

By carefully following each step, students can understand how the series is formed and how its sum is calculated. This methodical approach builds comprehension and confidence as they work through similar problems.

Mathematical Notation

Mathematical notation is a unique language designed to convey precise and concise information. Understanding this notation is crucial for solving problems in calculus and other math disciplines.
In this exercise, the summation notation \(\sum\) is used, which translates to adding up all specified terms of a sequence. Let's dissect the notation used:

• \(\sum_{i=6}^{8}\left(i^{2}+2\right)\): The symbol \(\sum\) denotes summation, which means to add a series of terms together.
• \(i=6\) is the starting point (lower limit) of the summation, while \(i=8\) is the ending point (upper limit).
• The expression \(i^2 + 2\) indicates the formula used to calculate each term of the sequence for each integer value of \(i\).

Understanding how to interpret and use these notations allows students to comprehend complex equations and calculations more easily and accurately. Good notation skills are like having the key to unlock and navigate the world of mathematical problems effortlessly.