/*! 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 50 Evaluate ar \(^{n}\) for the giv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 50

Evaluate ar \(^{n}\) for the given values of \(a, r,\) and \(n\) $$ a=3, r=2, n=4 $$

Short Answer

Expert verified
The value of ar^n for the given values is 24.

Step by step solution

01

- Understanding Geometric Sequences

This exercise requires evaluating the nth term of a geometric sequence. The general formula for the nth term of a geometric sequence is given by: \[ a_n = ar^{(n-1)} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
02

- Substituting Given Values

Using the given values \(a = 3\), \(r = 2\), and \(n = 4\), substitute these into the formula: \[ a_n = 3 \times 2^{(4-1)} \]
03

- Simplifying the Exponent

Simplify the expression inside the exponent first: \[ 4-1 = 3 \] Thus, the expression becomes: \[ a_n = 3 \times 2^3 \]
04

- Calculating the Power

Calculate \(2^3\): \[ 2^3 = 8 \]
05

- Completing the Calculation

Multiply the results to get the nth term: \[ a_n = 3 \times 8 = 24 \]

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.

nth term
The nth term represents the position of a term in a sequence. In a geometric sequence, it is given by the formula \( a_n = ar^{(n-1)} \). Each term after the first can be found by multiplying the previous term by the common ratio, r. For example, in the sequence 3, 6, 12, 24, ... the 4th term (when n=4) can be calculated using n, the term number. By plugging in n, you can determine the value of that term directly.
common ratio
The common ratio is a key aspect of a geometric sequence. It is the factor by which you multiply one term to get to the next. For the given example of \( a = 3, r = 2, n = 4 \), the common ratio r is 2. This means every term is twice the previous one. If you start with 3, multiplying by 2 gives 6, then 12, 24, and so on. The common ratio helps in understanding how quickly or slowly the terms in the sequence increase or decrease.
exponents
Exponents are used in geometric sequences to express the repeated multiplication of the common ratio. In our formula \( a_n = ar^{(n-1)} \), \( (n-1) \) is the exponent indicating how many times the common ratio is multiplied by itself. For instance, \( 2^{(4-1)} = 2^3 = 8 \). Exponents make it simpler to deal with large numbers in geometric sequences, which could otherwise be cumbersome to calculate.
geometric sequence formula
The geometric sequence formula provided is essential for calculating any term in the sequence directly. It is expressed as \( a_n = ar^{(n-1)} \). Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. By substituting the known values (\( a = 3, r = 2, n = 4 \)), you can find the 4th term: \( a_n = 3 \times 2^{(4-1)} = 3 \times 2^3 = 3 \times 8 = 24 \). This formula is a powerful tool in understanding and solving problems related to geometric sequences efficiently.

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

Solve each problem. Suppose that Cherian Mathew is offered a job at \(\$ 1600\) per month with a guaranteed increase of \(\$ 50\) every six months for 5 yr. What will Cherian's salary be at the end of that time?

Use a formula for \(S_{n}\) to evaluate each series. $$ \sum_{i=1}^{250} i $$

Find a general term for each geometric sequence. $$ -3, \frac{3}{2},-\frac{3}{4}, \ldots $$

Explain the basic difference between a sequence and a series.

Use a formula for \(S_{n}\) to evaluate each series. $$ \sum_{i=1}^{17}(3 i-1) $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.