/*! 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 11 What is the sum of all the even ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 11

What is the sum of all the even numbers from 0 to \(2 n,\) for any positive integer \(n ?\)

Short Answer

Expert verified
The sum of all even numbers from 0 to \(2n\) is \(n(n + 1)\).

Step by step solution

01

Identify the Even Numbers

The even numbers in the range from 0 to \(2n\) are: \(0, 2, 4, ..., 2n\). We need to sum these numbers.
02

Recognize the Arithmetic Sequence

Observe that the even numbers form an arithmetic sequence with the first term (a) as 0, the common difference (d) as 2, and the last term as \(2n\).
03

Determine the Number of Terms

To find the number of terms in the sequence, use the formula for the \(k\)-th term of an arithmetic sequence: \(a_k = a + (k-1)d\). Here, the last term \(2n = 0 + (k-1) \times 2\). Hence, \(2n = (k-1) \times 2 \ => k - 1 = n \ => k = n + 1\). So, there are \(n + 1\) terms.
04

Use the Sum Formula for Arithmetic Sequence

The sum \(S\) of the first \(k\) terms of an arithmetic sequence is given by the formula \(S = \frac{k}{2} \times (a + l)\), where \(a\) is the first term, \(l\) is the last term. Here, \(k = n + 1\), \(a = 0\), and \(l = 2n\). \ Hence, \(S = \frac{n + 1}{2} \times (0 + 2n) \ = \frac{n + 1}{2} \times 2n \ = (n + 1) \times n\). \ Therefore, \(S = n(n + 1)\).

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.

arithmetic sequences
An arithmetic sequence is a list of numbers where each number is obtained by adding a fixed value, called the common difference, to the previous number. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3 because each number is 3 more than the previous one.
Understanding the properties of arithmetic sequences helps in solving problems like summing even numbers.
sum of arithmetic series
The sum of an arithmetic series, which is just a fancy way of saying the sum of terms in an arithmetic sequence, can be found using a specific formula. If you have the first term (a), the last term (l), and the number of terms (k), you can use the formula: \[ S = \frac{k}{2} \times (a + l) \]
This formula tells you the sum (\textbf{S}) by multiplying the average of the first and last term by the number of terms, then dividing by 2.
even numbers
Even numbers are integers that are exactly divisible by 2, meaning there is no remainder when divided by 2. Examples include 0, 2, 4, 6, 8, and so on.
When looking at a range of even numbers, they naturally form an arithmetic sequence. For example, in the range from 0 to \(2n\), the even numbers are: 0, 2, 4, ..., \(2n\). The common difference (d) is 2.
Summing these numbers involves recognizing the arithmetic sequence and applying the sum formula for an arithmetic series.

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

The number of operations executed by algorithms \(A\) and \(B\) is \(40 n^{2}\) and \(2 n^{3},\) respectively. Determine \(n_{0}\) such that \(A\) is better than \(B\) for \(n \geq n_{0}\)

Given an \(n\) -element array \(X,\) Algorithm \(\mathrm{B}\) chooses \(\log n\) elements in \(X\) at random and executes an \(O(n)\) -time calculation for each. What is the worst-case running time of Algorithm B?

Algorithm \(A\) executes an \(O(\log n)\) -time computation for each entry of an \(n\) -element array. What is the worst-case running time of Algorithm \(A\) ?

Show that \(2^{n+1}\) is \(O\left(2^{n}\right)\).

Show that if \(d(n)\) is \(O(f(n))\) and \(e(n)\) is \(O(g(n)),\) then \(d(n)-e(n)\) is not necessarily \(O(f(n)-g(n))\)
See all solutions

Recommended explanations on Computer Science Textbooks

Blockchain Technology

Read Explanation

Data Representation in Computer Science

Read Explanation

Problem Solving Techniques

Read Explanation

Databases

Read Explanation

Fintech

Read Explanation

Game Design in Computer Science

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.