/*! 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 33 Compute each of these double sum... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 33

Compute each of these double sums. $$ \begin{array}{ll}{\text { a) } \sum_{i=1}^{2} \sum_{j=1}^{3}(i+j)} & {\text { b) } \sum_{i=0}^{2} \sum_{j=0}^{3}(2 i+3 j)} \\ {\text { c) } \sum_{i=1}^{3} \sum_{j=0}^{2} i} & {\text { d) } \sum_{i=0}^{2} \sum_{j=1}^{3} i j}\end{array} $$

Short Answer

Expert verified
a) 21, b) 78, c) 18, d) 18.

Step by step solution

01

- Sum for part (a)

Here we need to compute the double sum \[ \sum_{i=1}^{2} \sum_{j=1}^{3}(i+j) \]. First, calculate the inner sum for each value of i from 1 to 2.
02

- Sum for i = 1

\[ \sum_{j=1}^{3} (1+j) = (1+1) + (1+2) + (1+3) = 2 + 3 + 4 = 9 \]
03

- Sum for i = 2

\[ \sum_{j=1}^{3} (2+j) = (2+1) + (2+2) + (2+3) = 3 + 4 + 5 = 12 \]
04

- Total Sum for part (a)

Sum the results of the inner sums: \[ 9 + 12 = 21 \]
05

- Sum for part (b)

Compute the double sum \[ \sum_{i=0}^{2} \sum_{j=0}^{3}(2i+3j) \]. First, calculate the inner sum for each value of i from 0 to 2.
06

- Sum for i = 0

\[ \sum_{j=0}^{3} (2(0)+3j) = 3(0)+3(1)+3(2)+3(3) = 0 + 3 + 6 + 9 = 18 \]
07

- Sum for i = 1

\[ \sum_{j=0}^{3} (2(1)+3j) = 2 + 5 + 8 + 11 = 26 \]
08

- Sum for i = 2

\[ \sum_{j=0}^{3} (2(2)+3j) = 4 + 7 + 10 + 13 = 34 \]
09

- Total Sum for part (b)

Sum the results of the inner sums: \[ 18 + 26 + 34 = 78 \]
10

- Sum for part (c)

Compute the double sum \[ \sum_{i=1}^{3} \sum_{j=0}^{2} i. \] Here, the inner sum is constant as it only depends on i.
11

- Inner Sum Calculation for any i

\[ \sum_{j=0}^{2} i = i + i + i = 3i \]
12

- Total Sum for part (c)

\[ \sum_{i=1}^{3} 3i = 3(1) + 3(2) + 3(3) = 3 + 6 + 9 = 18 \]
13

- Sum for part (d)

Compute the double sum \[ \sum_{i=0}^{2} \sum_{j=1}^{3} ij. \] Calculate the inner sum for each value of i from 0 to 2.
14

- Sum for i = 0

\[ \sum_{j=1}^{3} (0 \cdot j) = 0 \]
15

- Sum for i = 1

\[ \sum_{j=1}^{3} (1 \cdot j) = 1 + 2 + 3 = 6 \]
16

- Sum for i = 2

\[ \sum_{j=1}^{3} (2 \cdot j) = 2 + 4 + 6 = 12 \]
17

- Total Sum for part (d)

Sum the results of the inner sums:\[ 0 + 6 + 12 = 18 \]

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.

Discrete Mathematics
Discrete mathematics studies structures that are fundamentally discrete rather than continuous. It involves mathematical objects such as integers, graphs, and statements in logic.
These structures do not vary smoothly but have distinct, separated values.
Because of its foundational role in computer science, discrete mathematics involves a lot of specific techniques and knowledge.
Nested Summation
Nested summation, also known as double sums, involves adding up numbers within multiple summation levels.
Each summation, typically represented by the summation symbol \(\sum \), operates within the outer summation.
For instance, in the exercise, we calculate inner sums for each value of an outer variable.
This concept is useful in various fields, from computer science algorithms to statistical calculations.
Step-by-Step Computation
Breaking down complex problems into smaller steps can make computations easier to understand.
Each step solves a part of the problem, contributing to the final answer.
  • **Step 1:** Identify the ranges for the outer and inner sums.
  • **Step 2:** Calculate the inner sum for each specific value of the outer variable.
  • **Step 3:** Sum the results obtained from Step 2 to get the final result.
Following these steps ensures clarity and accuracy in calculations.
Mathematical Notation
Mathematical notation is a system of symbols and signs used to represent numbers, functions, sets, and operations.
It is essential in exploring and communicating mathematical concepts clearly.
  • **Summation \(\sum \)** - A notation used to denote the addition of a sequence of numbers.
  • **Indexes** - Variables like \(i \) and \(j \) specify the range and increments of the summation.
  • **Formulas** - Expressions that describe the calculation rule, like \((i + j) \) or \(2i + 3j \).
Understanding these notations helps in interpreting complex equations and solving problems 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

What are the terms \(a_{0}, a_{1}, a_{2},\) and \(a_{3}\) of the sequence \(\left\\{a_{n}\right\\}\) where \(a_{n}\) equals $$ \begin{array}{ll}{\text { a) } 2^{n}+1 ?} & {\text { b) }(n+1)^{n+1} ?} \\\ {\text { c) }\lfloor n / 2\rfloor ?} & {\text { d) } \lfloor n / 2\rfloor+\lceil n / 2\rceil ?}\end{array} $$

Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. a) the function that assigns to each bit string the number of ones in the string minus the number of zeros in the string b) the function that assigns to each bit string twice the number of zeros in that string c) the function that assigns the number of bits left over when a bit string is split into bytes (which are blocks of 8 bits) d) the function that assigns to each positive integer the largest perfect square not exceeding this integer

Consider these functions from the set of students in a discrete mathematics class. Under what conditions is the function one-to-one if it assigns to a student his or her a) mobile phone number. b) student identification number. c) final grade in the class. d) home town.

For each of the following sets, determine whether 2 is an element of that set. $$ \begin{array}{l}{\text { a) }\\{x \in \mathbf{R} | x \text { is an integer greater than } 1\\}} \\ {\text { b) }\\{x \in \mathbf{R} | x \text { is the square of an integer }\\}} \\ {\text { c) }\\{2,\\{2\\}\\}} \\ {\text { e) }\\{\\{2\\},\\{2,\\{2\\}\\}\\}} & {\text { f) }\\{\\{2\\},\\{\\{2\\}\\}\\}}\end{array} $$

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the integers greater than 10 b) the odd negative integers c) the integers with absolute value less than \(1,000,000\) d) the real numbers between 0 and 2 e) the set \(A \times Z^{+}\) where \(A=\\{2,3\\}\) f) the integers that are multiples of 10
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

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.