Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 18: Problem 725

Mean of sequence \(1,2,4,8,16 \ldots \ldots .2^{n-1}\) is (a) \(\left[\left(2^{n}-1\right) / n\right]\) (b) \(\left[\left(2^{n-1}-1\right) /(n-1)\right]\) (c) \(\left[\left(2^{n}+1\right) / n\right]\) (d) \(\left[\left(2^{n}-1\right) /(n-1)\right]\)

Short Answer

Expert verified
The short answer is: \(\frac{2^n - 1}{n}\)

Step by step solution

01

The given sequence is \(1, 2, 4, 8, 16, ... , 2^{n-1}\). This is a geometric sequence with the first term \(a_1 = 1\) and a common ratio \(r = 2\). Let's summarize what we have: First term: \(a_1 = 1\) Common ratio: \(r = 2\) Number of terms: \(n\) #Step 2: Finding the sum of the geometric sequence#

In order to find the mean of the sequence, we need to find the sum of the sequence first. We use the formula for the sum \(S_n\) of a geometric sequence: \[S_n = \frac{a_1(r^n - 1)}{r - 1}\] Now, plug in the values for \(a_1\) and \(r\) that we found in Step 1: \[S_n = \frac{1(2^n - 1)}{2 - 1} = 2^n - 1\] #Step 3: Calculating the mean of the sequence#
02

Now that we have found the sum of the sequence, we can calculate the mean. The mean of a sequence is the sum divided by the number of terms. In this case: Mean = \(\frac{S_n}{n} = \frac{2^n - 1}{n}\) #Step 4: Comparing the calculated mean to the provided options#

We have calculated the mean of the sequence to be \(\frac{2^n - 1}{n}\). Now, let's compare this result to the provided options: (a) \(\left[\left(2^{n}-1\right) / n\right]\) (b) \(\left[\left(2^{n-1}-1\right) /(n-1)\right]\) (c) \(\left[\left(2^{n}+1\right) / n\right]\) (d) \(\left[\left(2^{n}-1\right) /(n-1)\right]\) Our calculated mean matches with option (a). Therefore, the correct answer is: (a) \(\left[\left(2^{n}-1\right) / n\right]\)

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A three digit number which is a multiple of 11 is chosen at random. The probability the number so chosen is also a multiple of 9 is (a) \((1 / 9)\) (b) \((2 / 9)\) (c) \((1 / 100)\) (d) \((9 / 100)\)

The A.M. of 9 terms is 15 . If one more term is added to this series then the A.M. becomes 16 . The value of added term is (a) 30 (b) 27 (c) 25 (d) 23

A dice is tossed until 1 comes. Then the probability that 1 comes in even number of trials is (a) \((5 / 11)\) (b) \((5 / 6)\) (c) \((6 / 11)\) (d) \((1 / 6)\)

10 balls are distributed among three boxes. Probability that the first box will contain 3 balls is (a) \(\left[\left({ }^{10} \mathrm{C}_{3} \times 2^{7}\right) / 3^{10}\right]\) (b) \(\left[\left({ }^{10} \mathrm{C}_{3} \times 2^{7}\right) / 10^{3}\right]\) (c) \(\left[\left({ }^{10} \mathrm{C}_{3} \cdot{ }^{7} \mathrm{C}_{2}\right) / 3^{10}\right]\) (d) \(\left[\left({ }^{10} \mathrm{P}_{3} \cdot 2^{7}\right) / 3^{10}\right]\)

The probability that a leap year will have 53 Sunday or 53 Monday is (a) \((2 / 7)\) (b) \((3 / 7)\) (c) \((4 / 7)\) (d) \((1 / 7)\)
See all solutions

Recommended explanations on English Textbooks

English Language Study

Read Explanation

Synthesis Essay

Read Explanation

Graphology

Read Explanation

Pragmatics

Read Explanation

Linguistic Terms

Read Explanation

Language Acquisition

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.

Sign-up for free