/*! 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 5 Evaluate the arithmetic series. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 5

Evaluate the arithmetic series. $$ 200+195+190+\cdots+75+70+65 $$

Short Answer

Expert verified
The arithmetic series given has a first term of \(a_1 = 200\), a common difference of \(d = -5\), and a total number of terms \(n = 28\). Using the arithmetic series sum formula, we can compute the sum as \(S_{28} = 28 \cdot \frac{200 + 65}{2} = 28 \cdot 132.5 = 3710\).

Step by step solution

01

Identify the first term, the common difference, and the number of terms

The first term of the series is 200. The common difference, which is the difference between each consecutive term, can be found by subtracting the second term from the first term or any term from its subsequent term: \[ 195 - 200 = -5 \] So, our common difference is -5. Next, we need to find the number of terms in the series. To do this, we will use the formula: \[ n = \frac{last\_term - first\_term}{common\_difference} + 1 \] Here, the last term is 65. Plugging in the values, we get: \[ n = \frac{65 - 200}{-5} + 1 \]
02

Find the number of terms in the series

Now, we will solve the equation for n: \[ n = \frac{-135}{-5} + 1 \] \[ n = 27 + 1 \] \[ n = 28 \] There are 28 terms in the series.
03

Use the arithmetic series sum formula

Now that we know the first term (\(a_1 = 200\)), the common difference (\(d = -5\)), and the number of terms (\(n = 28\)), we can use the formula for the sum of an arithmetic series: \[ S_n = n \cdot \frac{a_1 + a_n}{2} \] Here, \(a_n\) is the last term of the series (65). Plugging in the values, we get: \[ S_{28} = 28 \cdot \frac{200 + 65}{2} \]
04

Compute the sum

Now, we will compute the sum: \[ S_{28} = 28 \cdot \frac{265}{2} \] \[ S_{28} = 28 \cdot 132.5 \] \[ S_{28} = 3710 \] Thus, the sum of the arithmetic series is 3710.

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!

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 coefficient of \(t^{47}\) in the expansion of \((t+2)^{50}\).

Evaluate the geometric series. $$ \sum_{k=1}^{90} \frac{5}{7^{k}} $$

Evaluate \(\lim _{n \rightarrow \infty} n\left(\ln \left(3+\frac{1}{n}\right)-\ln 3\right)\).

Find the smallest integer \(n\) such that \(0.8^{n}<10^{-100}\).

Evaluate the arithmetic series. $$ \sum_{k=10}^{900}(3 k-2) $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.