Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 18: Problem 702

The mean of the series \(a, a+d, a+2 d \ldots \ldots . a+(2 n+1) d\) is (a) \(a+[(2 n+1) / 2] d\) (b) \(a+(n+1) d\) (c) \(a+(2 n+1) d\) (d) \(a+[(2 n-1) / 2] d\)

Short Answer

Expert verified
The mean of the given series is \(a + \frac{(2n+1)d}{2}\).

Step by step solution

01

Calculate the sum of the series

We have the series: \(a, a+d, a+2d, \ldots, a+(2n+1)d\). We can see that this is an arithmetic series with common difference d. The sum of an arithmetic series can be calculated using the formula: \[S = \frac{n}{2} (a_1 + a_n)\] where n is the number of terms, \(a_1\) is the first term and \(a_n\) is the last term. In our case, \(a_1 = a\) and \(a_n = a + (2n+1)d\). The total number of terms in the series is (2n+2). Now, let's find the sum of the series, S: \[S = \frac{2n+2}{2} (a + a + (2n+1)d)\]
02

Simplify the sum formula

Upon simplifying the formula for the sum, we get: \[S = (n+1)(2a+2n d + d)\] \[S = (n+1)(2a+(2n+1)d)\]
03

Calculate the mean of the series

Now we will find the mean of the series, which is the sum of the series divided by the total number of terms. As mentioned earlier, the total number of terms is (2n+2). Mean = \(\frac{S}{2n+2}\) Substitute the formula for S from Step 2: \[Mean = \frac{(n+1)(2a+(2n+1)d)}{2n+2}\]
04

Simplify the mean formula

Simplify the mean formula: \[Mean = \frac{(n+1)(2a+(2n+1)d)}{(n+1)(2)}\] \[Mean = \frac{2a+(2n+1)d}{2}\] Our final expression for the mean is: \[Mean = a + \frac{(2n+1)d}{2}\] Now we can see that this matches option (a): \(a + \frac{(2n+1)d}{2}\). So, the correct answer is (a).

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

There are two boxes. Box I contains 4 Red and 3 white balls. Box II contains 5 red and 2 white balls. Two balls are transferred from Box I to Box II. One ball is then drawn from box II randomly. What is the probability for that ball to be red? (a) \((43 / 63)\) (b) \((23 / 73)\) (c) \((34 / 63)\) (d) None

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

There are 4 addressed covers and 4 letters. If 4 letters are put in 4 covers randomly then the probability that not more than one letter is put in proper cover is (a) \((15 / 24)\) (b) \((7 / 24)\) (c) \((17 / 24)\) (d) \((7 / 17)\)

If \(p\) and \(q\) are chosen from \(\\{1,2,3,4,5,6,7,8,9,10\\}\) with replacement determine the probability that the roots of \(x^{2}+p x+q=0\) are real. (a) \(0.62\) (b) \(0.61\) (c) \(0.60\) (d) None

In any discrete series (when all values are not same) the relationship between M.D. about mean and S.D. is (a) M.D. = S.D. (b) M.D. \(\leq\) S.D. (c) M.D. < S.D. (d) M.D. \(\leq S . D\).
See all solutions

Recommended explanations on English Textbooks

History of English Language

Read Explanation

Key Concepts in Language and Linguistics

Read Explanation

Cues and Conventions

Read Explanation

Phonology

Read Explanation

Multiple Choice Questions

Read Explanation

Lexis and Semantics

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