/*! 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 35 Find the sum of the first five t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 35

Find the sum of the first five terms of the sequence. $$a_{n}=n^{2}-n$$

Short Answer

Expert verified
The sum of the first five terms is 40.

Step by step solution

01

Understand the Sequence Formula

The sequence formula given is \(a_n = n^2 - n\). This defines each term in the sequence based on its position \(n\).
02

Identify the Terms to Calculate

We need to find the first five terms of the sequence: \(a_1, a_2, a_3, a_4,\) and \(a_5\).
03

Calculate the First Term

Using the formula \(a_n = n^2 - n\), substitute \(n = 1\) to find \(a_1\): \[a_1 = 1^2 - 1 = 0\]
04

Calculate the Second Term

Substitute \(n = 2\) into the formula: \[a_2 = 2^2 - 2 = 4 - 2 = 2\]
05

Calculate the Third Term

Substitute \(n = 3\) into the formula: \[a_3 = 3^2 - 3 = 9 - 3 = 6\]
06

Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \[a_4 = 4^2 - 4 = 16 - 4 = 12\]
07

Calculate the Fifth Term

Substitute \(n = 5\) into the formula: \[a_5 = 5^2 - 5 = 25 - 5 = 20\]
08

Sum the First Five Terms

Add up all the terms we have calculated: \[0 + 2 + 6 + 12 + 20 = 40\]

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.

Sequence Formula
A sequence formula helps define the rule that generates each term in a sequence based on their position in the sequence. For a sequence, each term is represented by a formula that involves an arithmetic operation.
\( a_n = n^2 - n \) is such a sequence formula identifying any given term by placing the term number \( n \) into the formula.
For example:
  • \( a_1 = 1^2 - 1 = 0 \)
  • \( a_2 = 2^2 - 2 = 2 \)
  • \( a_3 = 3^2 - 3 = 6 \)
By substituting different values of \( n \), one can compute each term's value, effectively understanding the behavior and pattern of the sequence as a whole. Each term follows that specific pattern defined by the formula, forming a structure for the sequence.
Arithmetic Sequences
Arithmetic sequences are sequences where consecutive terms have a constant difference between them. This difference is called the "common difference." However, not all sequences are arithmetic. In our example, the sequence defined by \( a_n = n^2 - n \) is not arithmetic.
Arithmetic sequences are structured as follows:
  • They can be predicted using the formula \( a_n = a_1 + (n-1) \, d \), where \( d \) is the common difference.
  • They have a linear pattern with terms evenly spaced in addition.
  • Each term can affect the next by the addition of a constant difference.
Understanding whether a sequence is arithmetic or not is useful, as it helps predict future terms quickly and identify necessary calculations for sums.
Series and Sums
When dealing with sequences, a series represents the sum of a sequence's terms that have been defined by a specific formula or pattern. The term "sum" usually refers to adding all these terms together to find a total. This process is particularly useful when trying to find the cumulative value of parts of a large sequence.
To find the sum of the first five terms like in our exercise:
  • Calculate each of the first five terms using the given formula.
  • Apply summation to these results: \( 0 + 2 + 6 + 12 + 20 = 40 \).
Summation helps in understanding not just individual elements but the cumulative whole that they create. The calculation uses the sequence formula repeatedly to find each component before adding them together to find the desired result.

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

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$\sqrt{2}\left(\cos \frac{1}{3} \pi+i \sin \frac{1}{3} \pi\right) \times \sqrt{2}\left(\cos \frac{4}{3} \pi+i \sin \frac{4}{3} \pi\right)$$

Convert from rectangular to trigonometric form. (In each case, choose an argument \theta such that \(0 \leq \theta<2 \pi .)\) $$\frac{1}{4} \sqrt{3}-\frac{1}{4} i$$

Suppose that \(a, b,\) and \(c\) are three positive numbers with \(a>c>2 b .\) If \(1 / a, 1 / b,\) and \(1 / c\) are consecutive terms in an arithmetic sequence, show that $$\ln (a+c)+\ln (a-2 b+c)=2 \ln (a-c)$$

Let \(a_{1}=1 /(1+\sqrt{2}), a_{2}=-1,\) and \(a_{3}=1 /(1-\sqrt{2})\) (a) Show that \(a_{2}-a_{1}=a_{3}-a_{2}\) (b) Find the sum of the first six terms in the arithmetic sequence $$\frac{1}{1+\sqrt{2}}, \quad-1, \quad \frac{1}{1-\sqrt{2}}, \dots$$

Rewrite the sums using sigma notation. $$x+2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+6 x^{6}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.