All Courses
Explore Courses UPSC Exam UPSC Test Series Free Notes EduRev Infinity Get the App Login Join for Free
Sign in Open App
GMAT Exam  >  GMAT Questions  >  When positive integer y is added to each of t... Start Learning for Free
SaveJoin the Discussion on the App
When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?
I. If the median of the resulting numbers is   then n is odd
II. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbers
III. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I, II and III
  • e)
    None of the above
Correct answer is option 'B'. Can you explain this answer?
FREE This question is part of
Download PDF Attempt Test
Download PDF Attempt this Test
Verified Answer
When positive integer y is added to each of the first n non-negative i...
Given:
  • The list of first n non-negative integers: {0, 1, 2, 3, . . . , n – 1}
  • Positive integer y is added to each integer in this list: {0 + y, 1 + y, 2 + y, . . . n – 1 + y}
    • = {y, y + 1, y + 2, . . . , y + n – 1}
To Find: Which of the 3 statements is/are true?
Approach:
  1. Since these 3 statements deal with:
    • Mean of the first n positive integers
    • Mean of the resulting numbers
    • And, Median of the resulting numbers,
We will first find the expressions for these 3 quantities.
2. Then, we’ll evaluate the 3 statements one by one to determine which is/are true for all values of y and n
Working out:
  • Finding the expressions for the 3 quantities featured in Statements I – III
     
    • Finding Mean of the first n positive integers
      • Sum of first n positive integers = 
      • So, the mean of the first n positive integers =  
        • (n+2/2)
  • Finding Mean of the Resulting Numbers
    • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n – 1}
      • These numbers form an increasing arithmetic sequence of n terms.
        • First term of the sequence = y
        • Last term of the sequence = y + n -1
        • So, the sum of these numbers =
 
  • Finding Median of the Resulting Numbers
    • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n – 1}
    • The total number of elements in this set is (y + n – 1) – y + 1 = n
      • These numbers form an increasing arithmetic sequence of n terms.
      • Now, in an ordered list that has:
        • An even number of elements (say 4 elements), the median of the list is equal to the average of the middle 2 elements of the list
        • An odd number of elements (Say 5 elements), the median of the list is equal to the middle element in the list
 
  • Case 1: If n is odd,
  • Then, Median = the middle element in the list of resulting numbers
  • The first term in the list is y + 0 and the last term is y +(n – 1)
  • So, the Median = 
  • (Note: If the above expression for the Median is not intuitive to you, you can arrive at it by taking a few easy values of n. For example:
    • If n = 3, the list is {y, y + 1, y + 2}. So, the median = y + 1
    • If n = 5, the list is {y, y + 1. y + 2, y + 3, y + 4}. So, the median = y + 2
    • Similarly, if n = 7, the list goes from y to y + 6 and the median = y + 3
    • From these examples, the pattern for how the value of Median changes with n becomes easy to see)
  • Case 2: If n is even,
    • This means, the median of the list is equal to the​
 
  • Evaluating Statement I
    • If the median of the resulting numbers is then n is odd
    • In our calculation of the Median of the Resulting Numbers, observe that the median is always equal to  , whether n is even or odd.
    • Therefore, Statement I is not correct
  • Evaluating Statement II
    • The arithmetic mean of the resulting numbers is equal to the median of the resulting numbers
      • From our calculations of the Mean and Median of the Resulting Numbers, we see that:
        • Mean of the Resulting numbers =
        • Median =
    • So, Statement II is indeed true.
  • Evaluating Statement III
    • The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.
    • From our calculations above, we see that:
      • Mean of the Resulting numbers = 
      • Mean of the first n positive integers 
    • Note that  is not equal to  . Therefore, it is wrong to say that Mean of the Resulting Numbers is y units greater than the Mean of the first n positive integers.
    • So, Statement III is not true.
 
  • Getting to the answer
    • Of the 3 statements, we see that only Statement II is true.
Looking at the answer choices, we see that the correct answer is Option B
View all questions of this test
Most Upvoted Answer
When positive integer y is added to each of the first n non-negative i...
Given:
  • The list of first n non-negative integers: {0, 1, 2, 3, . . . , n – 1}
  • Positive integer y is added to each integer in this list: {0 + y, 1 + y, 2 + y, . . . n – 1 + y}
    • = {y, y + 1, y + 2, . . . , y + n – 1}
To Find: Which of the 3 statements is/are true?
Approach:
  1. Since these 3 statements deal with:
    • Mean of the first n positive integers
    • Mean of the resulting numbers
    • And, Median of the resulting numbers,
We will first find the expressions for these 3 quantities.
2. Then, we’ll evaluate the 3 statements one by one to determine which is/are true for all values of y and n
Working out:
  • Finding the expressions for the 3 quantities featured in Statements I – III
     
    • Finding Mean of the first n positive integers
      • Sum of first n positive integers = 
      • So, the mean of the first n positive integers =  
        • (n+2/2)
  • Finding Mean of the Resulting Numbers
    • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n – 1}
      • These numbers form an increasing arithmetic sequence of n terms.
        • First term of the sequence = y
        • Last term of the sequence = y + n -1
        • So, the sum of these numbers =
 
  • Finding Median of the Resulting Numbers
    • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n – 1}
    • The total number of elements in this set is (y + n – 1) – y + 1 = n
      • These numbers form an increasing arithmetic sequence of n terms.
      • Now, in an ordered list that has:
        • An even number of elements (say 4 elements), the median of the list is equal to the average of the middle 2 elements of the list
        • An odd number of elements (Say 5 elements), the median of the list is equal to the middle element in the list
 
  • Case 1: If n is odd,
  • Then, Median = the middle element in the list of resulting numbers
  • The first term in the list is y + 0 and the last term is y +(n – 1)
  • So, the Median = 
  • (Note: If the above expression for the Median is not intuitive to you, you can arrive at it by taking a few easy values of n. For example:
    • If n = 3, the list is {y, y + 1, y + 2}. So, the median = y + 1
    • If n = 5, the list is {y, y + 1. y + 2, y + 3, y + 4}. So, the median = y + 2
    • Similarly, if n = 7, the list goes from y to y + 6 and the median = y + 3
    • From these examples, the pattern for how the value of Median changes with n becomes easy to see)
  • Case 2: If n is even,
    • This means, the median of the list is equal to the​
 
  • Evaluating Statement I
    • If the median of the resulting numbers is then n is odd
    • In our calculation of the Median of the Resulting Numbers, observe that the median is always equal to  , whether n is even or odd.
    • Therefore, Statement I is not correct
  • Evaluating Statement II
    • The arithmetic mean of the resulting numbers is equal to the median of the resulting numbers
      • From our calculations of the Mean and Median of the Resulting Numbers, we see that:
        • Mean of the Resulting numbers =
        • Median =
    • So, Statement II is indeed true.
  • Evaluating Statement III
    • The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.
    • From our calculations above, we see that:
      • Mean of the Resulting numbers = 
      • Mean of the first n positive integers 
    • Note that  is not equal to  . Therefore, it is wrong to say that Mean of the Resulting Numbers is y units greater than the Mean of the first n positive integers.
    • So, Statement III is not true.
 
  • Getting to the answer
    • Of the 3 statements, we see that only Statement II is true.
Looking at the answer choices, we see that the correct answer is Option B
Free Test
FREE
Start Free Test
Start Free Test
Community Answer
When positive integer y is added to each of the first n non-negative i...
Given:
  • The list of first n non-negative integers: {0, 1, 2, 3, . . . , n – 1}
  • Positive integer y is added to each integer in this list: {0 + y, 1 + y, 2 + y, . . . n – 1 + y}
    • = {y, y + 1, y + 2, . . . , y + n – 1}
To Find: Which of the 3 statements is/are true?
Approach:
  1. Since these 3 statements deal with:
    • Mean of the first n positive integers
    • Mean of the resulting numbers
    • And, Median of the resulting numbers,
We will first find the expressions for these 3 quantities.
2. Then, we’ll evaluate the 3 statements one by one to determine which is/are true for all values of y and n
Working out:
  • Finding the expressions for the 3 quantities featured in Statements I – III
     
    • Finding Mean of the first n positive integers
      • Sum of first n positive integers = 
      • So, the mean of the first n positive integers =  
        • (n+2/2)
  • Finding Mean of the Resulting Numbers
    • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n – 1}
      • These numbers form an increasing arithmetic sequence of n terms.
        • First term of the sequence = y
        • Last term of the sequence = y + n -1
        • So, the sum of these numbers =
 
  • Finding Median of the Resulting Numbers
    • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n – 1}
    • The total number of elements in this set is (y + n – 1) – y + 1 = n
      • These numbers form an increasing arithmetic sequence of n terms.
      • Now, in an ordered list that has:
        • An even number of elements (say 4 elements), the median of the list is equal to the average of the middle 2 elements of the list
        • An odd number of elements (Say 5 elements), the median of the list is equal to the middle element in the list
 
  • Case 1: If n is odd,
  • Then, Median = the middle element in the list of resulting numbers
  • The first term in the list is y + 0 and the last term is y +(n – 1)
  • So, the Median = 
  • (Note: If the above expression for the Median is not intuitive to you, you can arrive at it by taking a few easy values of n. For example:
    • If n = 3, the list is {y, y + 1, y + 2}. So, the median = y + 1
    • If n = 5, the list is {y, y + 1. y + 2, y + 3, y + 4}. So, the median = y + 2
    • Similarly, if n = 7, the list goes from y to y + 6 and the median = y + 3
    • From these examples, the pattern for how the value of Median changes with n becomes easy to see)
  • Case 2: If n is even,
    • This means, the median of the list is equal to the​
 
  • Evaluating Statement I
    • If the median of the resulting numbers is then n is odd
    • In our calculation of the Median of the Resulting Numbers, observe that the median is always equal to  , whether n is even or odd.
    • Therefore, Statement I is not correct
  • Evaluating Statement II
    • The arithmetic mean of the resulting numbers is equal to the median of the resulting numbers
      • From our calculations of the Mean and Median of the Resulting Numbers, we see that:
        • Mean of the Resulting numbers =
        • Median =
    • So, Statement II is indeed true.
  • Evaluating Statement III
    • The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.
    • From our calculations above, we see that:
      • Mean of the Resulting numbers = 
      • Mean of the first n positive integers 
    • Note that  is not equal to  . Therefore, it is wrong to say that Mean of the Resulting Numbers is y units greater than the Mean of the first n positive integers.
    • So, Statement III is not true.
 
  • Getting to the answer
    • Of the 3 statements, we see that only Statement II is true.
Looking at the answer choices, we see that the correct answer is Option B
Attention GMAT Students!
To make sure you are not studying endlessly, EduRev has designed GMAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in GMAT.
< Previous Question Next Question >
Explore Courses for GMAT exam

Similar GMAT Doubts

Top Courses for GMATView all

Top Courses for GMAT

When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
Question Description
When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer?.
Solutions for When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT. Download more important topics, notes, lectures and mock test series for GMAT Exam by signing up for free.
Here you can find the meaning of When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice When positive integer y is added to each of the first n non-negative integers, which of the following statements is true?I. If the median of the resulting numbers is then n is oddII. The arithmetic mean of the resulting numbers is equal to the median of the resulting numbersIII. The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice GMAT tests.
Explore Courses for GMAT exam

Top Courses for GMAT

Explore Courses

Suggested Free Tests

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Get App
© EduRev
Education Revolution
Signup to see your scores go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup