/*! 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 73 Computer animations: The animati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 73

Computer animations: The animation on a new computer game initially allows the hero of the game to jump a (screen) distance of 10 in. over booby traps and obstacles. Each successive jump is limited to \(\frac{3}{4}\) in. less than the previous one. Find (a) the length of the seventh jump and (b) the total distance covered after seven jumps.

Short Answer

Expert verified
(a) 5.5 inches; (b) 54.25 inches.

Step by step solution

01

Understand the Problem

We need to find the length of the seventh jump and the total distance covered after seven jumps. The initial jump is 10 inches, and each successive jump is 0.75 inches less.
02

Identify the Pattern of the Sequence

This is an arithmetic sequence where each term is 0.75 inches less than the previous term. The first term \(a_1\) is 10 inches, and the common difference \(d\) is -0.75 inches.
03

Find the Seventh Jump

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). Plug \(n=7\), \(a_1=10\), and \(d=-0.75\) into the formula: \[ a_7 = 10 + (7-1) \cdot (-0.75) = 10 - 4.5 = 5.5 \text{ inches} \] So, the length of the seventh jump is 5.5 inches.
04

Calculate Total Distance for Seven Jumps

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by \(S_n = \frac{n}{2} (a_1 + a_n)\). We know \(a_1 = 10\), \(a_7 = 5.5\), and \(n = 7\). \[ S_7 = \frac{7}{2} (10 + 5.5) = \frac{7}{2} \cdot 15.5 = 54.25 \text{ inches} \] Thus, the total distance covered after seven jumps is 54.25 inches.

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.

Arithmetic Sequence Formula
An arithmetic sequence is a sequence of numbers where the difference between any two successive terms is constant. This constant difference is known as the 'common difference'. The general formula to find the nth term of an arithmetic sequence, denoted as \( a_n \), is expressed as: \[ a_n = a_1 + (n-1) \cdot d \]This formula allows you to determine any term in the sequence, provided you know:
  • the first term \( a_1 \)
  • the common difference \( d \)
  • the term position \( n \)
Understanding this formula is key to solving problems related to arithmetic sequences, as it helps calculate not just specific terms but also explore the pattern within the entire sequence.
Common Difference
The common difference is a crucial element of an arithmetic sequence. It indicates how much you need to add or subtract to get from one term to the next. In mathematical terms, the common difference \( d \) can be calculated by:
  • subtracting any term from the subsequent term, such as \( a_2 - a_1 \)
In our example, with the hero's jump distances:
  • The first jump is 10 inches.
  • The second jump is \(10 - 0.75 = 9.25 \) inches.
The common difference here is \(-0.75\) inches. This negative difference indicates that the jump lengths are decreasing with each subsequent jump. Recognizing and applying the common difference helps us understand how the sequence progresses.
Nth Term Calculation
The 'nth term calculation' in an arithmetic sequence helps to find a specific term鈥檚 value given its position. Using our example, we calculated the seventh jump with the nth term formula:\[ a_n = a_1 + (n-1) \cdot d \]For the hero's seventh jump:
  • \( a_1 = 10 \) inches, the first term.
  • \( d = -0.75 \) inches, the common difference.
  • \( n = 7 \) for the seventh term.
By substituting these values into the formula, we calculated:\[ a_7 = 10 + (7-1) \cdot (-0.75) = 5.5 \text{ inches} \]So, the length of the seventh jump is 5.5 inches. This method provides a straightforward way to find any given term in an arithmetic sequence.
Sum of Arithmetic Sequence
The sum of an arithmetic sequence is a useful calculation that provides the total of multiple terms in the sequence. The general formula used is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]This formula requires:
  • \( n \), the number of terms.
  • \( a_1 \), the first term.
  • \( a_n \), the nth term.
In the example of the hero's jumps, to find the total distance covered after seven jumps, we used:
  • \( n = 7 \)
  • \( a_1 = 10 \)
  • \( a_7 = 5.5 \)
Plugging these into the sum formula:\[ S_7 = \frac{7}{2} (10 + 5.5) = 54.25 \text{ inches} \]This result shows the total distance covered by all seven jumps. Understanding the sum formula aids in calculating overall values in scenarios involving arithmetic sequences.

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

Determine if the sequence given is geometric. If yes, name the common ratio. If not, try to determine the pattern that forms the sequence. $$25,10,4, \frac{8}{5}, \dots$$

A number is called a "perfect number" if the sum of its proper factors is equal to the number itself. Six is the first perfect number since the sum of its proper factors is six: \(1+2+3=6 .\) Twenty-eight is the second since: \(1+2+4+7+14=28 .\) A young child is given a box containing eight wooden blocks with the following numbers (one per block) printed on them: four 3 's, two 5 's, one \(0,\) and one \(6 .\) What is the probability she draws the eight blocks in order and forms the fifth perfect number: \(33,550,336 ?\)

"Double-six" dominos is a game played with the 28 numbered tiles shown in the diagram. (Diagram Can't Copy) The 28 dominos are placed in a bag, shuffled, and then one domino is randomly drawn. What is the probability the total number of dots on the domino a. is three or less b. is greater than three c. does not have a blank half d. is not a "double" (both sides the same)

Pinochle is a card game played with a deck of 48 cards consisting of 2 Aces, 2 Kings, 2 Queens, 2 Jacks, 2 Tens, and 2 Nines in each of the four standard suits [hearts , diamonds, spades, and clubs]. If one card is drawn at random from this deck, what is the probability of a. drawing an Ace b. drawing a club c. drawing a red card d. drawing a face card (Jack, Queen, King)

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed. In any binomial expansion, there is always _________ more term than the power applied.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

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.