/*! 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 71 Find the partial sum without usi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 71

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{50} n$$

Short Answer

Expert verified
The partial sum of the first 50 natural numbers is 1275.

Step by step solution

01

Understand the task

We need to find the partial sum of the first 50 natural numbers. This is an arithmetic series where the first term (a) is 1, the last term (l) is 50 and the number of terms (n) is 50.
02

Apply the formula

The formula for the sum of an arithmetic series is \( S = \frac{n}{2}*(a + l) \). Here, \(a = 1\), \(l = 50\), and \(n = 50\).
03

Calculate the Sum

Substitute the values into the formula to get the sum: \( S = \frac{50}{2}*(1 + 50) \) which simplifies to \( S = 25*(51) = 1275\).

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.

Partial Sum
A partial sum refers to the sum of a specific part of a sequence or series, rather than the entire series. Think of it as adding up a sequence but stopping at a certain point. For example, if we want the partial sum of natural numbers up to 50, we only add numbers from 1 to 50, rather than continuing indefinitely.
Partial sums are crucial in arithmetic series because they allow us to find the sum of a sequence up to a specific term without listing all the numbers. This makes solving problems more efficient.
  • Focus: Specific part of the series
  • Example: Sum from 1 to 50
  • Benefit: Simplifies complex calculations
Natural Numbers
Natural numbers are the set of positive integers beginning from 1 and moving upwards without end: 1, 2, 3, and so on. They are fundamental in mathematics and are used for counting and ordering.
Natural numbers include just the positive numbers that you would naturally use to count objects. They do not include zero, fractions, or negative numbers.
  • Sequence: 1, 2, 3, 4, ...
  • Role in Arithmetic: Used in basic operations
  • Properties: Infinite and positive
Formula for Sum of Arithmetic Series
The formula for the sum of an arithmetic series is a direct way to find the sum of numbers evenly spaced out in a sequence. An arithmetic series has terms that increase or decrease by the same amount, known as the common difference.
For a series, the sum can be found using: \[ S = \frac{n}{2}(a + l) \] where:
  • \( n \): Number of terms
  • \( a \): First term
  • \( l \): Last term
This formula helps in quickly finding the total of any arithmetic sequence without having to add each term individually. It works effectively for sequences big or small. For instance, this formula was used to find the sum of the first 50 natural numbers, where the result was 1275.

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

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}$$

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{1}{(n+1) !}$$

Use the Binomial Theorem to expand and simplify the expression. \((x+y)^{5}\)

Write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\) th term of the sequence as a function of \(n .\) (Assume \(n\) begins with 1.) $$a_{1}=25, a_{k+1}=a_{k}-5$$

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{250}(1000-n)$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.