/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 Find each sum. See Example 9. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 65

Find each sum. See Example 9. $$ \sum_{k=3}^{4}\left(k^{2}+3\right) $$

Short Answer

Expert verified
The sum is 31.

Step by step solution

01

Understand the Summation

The given sum is \( \sum_{k=3}^{4}(k^2 + 3) \). This notation tells us to evaluate the expression \( k^2 + 3 \) for each integer \( k \) from 3 to 4, inclusive, and then sum those values.
02

Evaluate the Expression for Each k

First, substitute \( k = 3 \) into the expression: \((3^2 + 3) = 9 + 3 = 12\). Next, substitute \( k = 4 \) into the expression: \((4^2 + 3) = 16 + 3 = 19\).
03

Calculate the Sum

Now, add the results from each substitution: \( 12 + 19 = 31 \). This is the sum of the series for the given \( k \) values.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Expressions
Understanding how to evaluate expressions is an essential skill in algebra. In the context of summation notation, evaluating expressions involves substituting specific values into a mathematical equation. Imagine you have an expression like \(k^2 + 3\). Your task is to evaluate this expression for different values of \(k\). For example:
  • If \(k = 3\), substitute 3 into the expression: \(3^2 + 3\), which simplifies to \(9 + 3 = 12\).

  • If \(k = 4\), substitute 4 into the expression: \(4^2 + 3\), which simplifies to \(16 + 3 = 19\).
Evaluating expressions often involves careful substitution and arithmetic. Each result contributes to the final outcome of the summation. By breaking down the steps, you gain clarity and prevent errors that might arise from handling complex calculations.
Integer Values
Dealing with integer values makes the process of summation straightforward, especially in sequences and series. Integers are whole numbers without any fractional parts, including negative numbers, zero, and positive numbers.In our given exercise, the integer values of \(k\) range from 3 to 4. Integer inputs give exact outputs, making calculations more manageable:
  • The selected integer for the first evaluation is \(k=3\).

  • The next integer in the series is \(k=4\).
Using integers ensures that each step in the summation is clear-cut and involves simple arithmetic. This simplicity helps focus on the process rather than the arithmetic complexity usually encountered with non-integers like fractions or decimals.
Step-by-Step Solution
Approaching any mathematical problem with a step-by-step solution is a helpful strategy. It provides a roadmap for solving the problem systematically.For the sum \(\sum_{k=3}^{4}(k^2 + 3)\), we solve it in clear stages:
  • Understand the Task: Recognize that you need to find the value of the expression for each \(k\) within the set range.

  • Substitute Values: Calculate the expression \(k^2 + 3\) for \(k=3\) and for \(k=4\).

  • Compute the Sum: Add the results. Here, the sum is \(12 + 19\), giving a final sum of \(31\).
This methodical approach reduces the chance of mistakes and increases understanding of each calculation's role in the larger picture. As you gain more confidence, you'll find these steps second nature in solving complex summations efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the first six terms of the arithmetic sequence if the common difference is \(-3\) and the ninth term is \(10 .\)

Find the 40 th term of the arithmetic sequence with a second term of 6 and a fourth term of 16.

Find the sum of each infinite geometric series, if possible. $$ 8+4+2+\cdots $$

Find each sum. See Example 9. $$ \sum_{k=2}^{5}(3 k) $$

Find the sum of each infinite geometric series, if possible. $$ 45+15+5+\cdots $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.