/*! 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 51 Expand the partial sum and find ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 51

Expand the partial sum and find its value. \(\sum_{i=1}^{6}(2 i+3)\)

Short Answer

Expert verified
The value of the partial sum is 60.

Step by step solution

01

Identify the Sum

The given sum is a partial sum \( \sum_{i=1}^{6}(2i+3) \). This means summing the expression \( 2i + 3 \) for each integer value of \ i \ from 1 to 6.
02

Calculate Each Term

Evaluate the expression \( 2i + 3 \) for each \ i \ value from 1 to 6: \( i = 1: 2(1) + 3 = 5 i = 2: 2(2) + 3 = 7 i = 3: 2(3) + 3 = 9 i = 4: 2(4) + 3 = 11 i = 5: 2(5) + 3 = 13 i = 6: 2(6) + 3 = 15 \).
03

Sum the Terms

Add the calculated values together: \( 5 + 7 + 9 + 11 + 13 + 15 = 60 \).

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.

Summation Notation
Summation notation is a convenient way to represent the sum of a sequence of terms. It is often used in mathematics to compactly write the sum of series. The notation \( \sum_{i=1}^{6}(2i+3) \) means that we are summing the expression \( 2i + 3 \) where \( i \) varies from 1 to 6. This allows us to easily see which terms are included in the sum. For example:
\[ \sum_{i=1}^{6}(2i+3) = (2(1)+3) + (2(2)+3) + (2(3)+3) + (2(4)+3) + (2(5)+3) + (2(6)+3) \]
Summation notation helps students efficiently solve problems and is essential for more advanced mathematical concepts.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. In our case, the sequence formed by the expression \( 2i + 3 \) is an arithmetic sequence. By evaluating the expression for each \( i \) from 1 to 6, we obtained the terms: 5, 7, 9, 11, 13, and 15. These terms are evenly spaced by a common difference of 2, which you can confirm by subtracting any term from the next one:
\( 7 - 5 = 2, \ 9 - 7 = 2, \ 11 - 9 = 2, \ 13 - 11 = 2, \ 15 - 13 = 2 \)
Arithmetic sequences are fundamental in understanding series and sums in mathematics.
Series Expansion
Series expansion is the method of expressing a series in its expanded form. By expanding the series \( \sum_{i=1}^{6}(2i+3) \) we clearly see each individual term that we are summing:
\[ 5 + 7 + 9 + 11 + 13 + 15 \]
Expanding the series helps students understand the structure of the sequence and compute the sum step by step. By simplifying, we can also understand patterns and relationships between terms. In our example, the expanded form made it easy to add each term to get the total sum of 60.
Algebraic Expression
An algebraic expression is a combination of numbers, variables, and arithmetic operations. The given sum \( \sum_{i=1}^{6}(2i+3) \) is an example of an algebraic expression. Here, \( 2i + 3 \) includes a variable \( i \) and constants 2 and 3 combined using addition and multiplication.
Evaluating this algebraic expression for different values of \( i \) helped us find the terms of the series. Understanding algebraic expressions is crucial because they form the foundation for solving equations, finding patterns, and formulating mathematical models.

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 indicated term of a sequence where the first term and the common ratio is given. . Find \(a_{11}\) given \(a_{1}=8\) and \(r=3\).

Find the indicated term in the expansion of the binomial. Seventh term of \((x-y)^{11}\)

Write the first five terms of each geometric sequence with the given first term and common ratio. \(a_{1}=4\) and \(r=3\)

Find the sum of each infinite geometric series. \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\ldots\)

Determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio. \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \ldots\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.