/*! 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 70 Find the sum for each series. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 70

Find the sum for each series. $$\sum_{i=1}^{6}\left(i^{2}+2 i^{3}\right)$$

Short Answer

Expert verified
The sum of the series is 973.

Step by step solution

01

Understand the Series Structure

The series given is \( \sum_{i=1}^{6} (i^2 + 2i^3) \). This means that for each integer \( i \) from 1 to 6, you calculate \( i^2 + 2i^3 \) and then sum all these values.
02

Calculate Each Term

Calculate each term in the series by substituting values of \( i \) from 1 to 6:- For \( i = 1 \), term is \( 1^2 + 2 \times 1^3 = 1 + 2 = 3 \)- For \( i = 2 \), term is \( 2^2 + 2 \times 2^3 = 4 + 16 = 20 \)- For \( i = 3 \), term is \( 3^2 + 2 \times 3^3 = 9 + 54 = 63 \)- For \( i = 4 \), term is \( 4^2 + 2 \times 4^3 = 16 + 128 = 144 \)- For \( i = 5 \), term is \( 5^2 + 2 \times 5^3 = 25 + 250 = 275 \)- For \( i = 6 \), term is \( 6^2 + 2 \times 6^3 = 36 + 432 = 468 \)
03

Sum All Terms

Add all the calculated terms together: - Sum = 3 + 20 + 63 + 144 + 275 + 468 = 973.

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 compact way to represent the addition of a sequence of numbers. It is typically written using the Greek letter sigma (\( \Sigma \)). In this context, the expression \( \sum_{i=1}^{6} (i^2 + 2i^3) \) means you start by setting \( i = 1 \) and go up to \( i = 6 \).
For each value of \( i \), you compute \( i^2 + 2i^3 \) and sum these results. Using summation notation helps keep complex calculations organized and easily readable, which simplifies working with series calculations.
  • \(i = 1 \to i^2 + 2i^3 = 1^2 + 2 \times 1^3 = 3\)
  • \(i = 2 \to i^2 + 2i^3 = 2^2 + 2 \times 2^3 = 20\)
  • Continue similarly for each \(i\) up to 6.
Remember, the numbers represent terms that form the series, offering a method to systematically add them.
Polynomial Expressions
A polynomial expression is an algebraic expression that consists of variables and coefficients, structured in terms of powers and sums. In the problem given, the expression inside the summation notation \((i^2 + 2i^3)\) is a polynomial in terms of \(i\).
Polynomials are classified based on their degree, which is the highest power of the variable in the expression. For instance, in \(i^2 + 2i^3\):
  • The term \(i^2\) is a polynomial of degree 2.
  • The term \(2i^3\) is of degree 3, making it the dominant term of the polynomial expression.
These polynomial expressions, especially in series, can represent more complex relationships and increase the variety of sequences that one can sum, making them versatile for many mathematical applications.
Series Calculation
Series calculation involves summing a sequence of numbers or terms that follow a specific pattern or formula. The series in this exercise is expressed as \( \sum_{i=1}^{6} (i^2 + 2i^3) \), where each term is calculated based on the specified formula before summing them together.
The process breaks down into manageable steps:
  • Substitute values: Insert each integer from the defined range into the polynomial expression \((i^2 + 2i^3)\).
  • Calculate each term: Compute values for each individual \(i\): \(3, 20, 63, 144, 275, 468\).
  • Add the terms: Summing these results: \(3 + 20 + 63 + 144 + 275 + 468\) gives the total \(973\).
Series calculations can be found in statistics, calculus, and financial mathematics, showing how crucial understanding this process is for broader applications.

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 \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{12}=-108, a_{12}=-19$$

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n}(k+2) \leq 52$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) Exactly 3 girls and 2 boys

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) Exactly 2 girls and 3 boys

Find the sum of each series. $$\sum_{i=1}^{1000} i$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.