Class 11 Exam > Class 11 Questions > if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N...
Start Learning for Free
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y
Verified Answer
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subse...
All we need to prove is that 4n - 3n - 1 is divisible by 9
Let n = 1
41 - 3(1) - 1 = 0 which is divisible by 9.
Suppose the hypothesis is true for n and consider n + 1.
4n+1 - 3(n + 1) - 1 =
2 - 3n - 3 - 1 =
2(2n)22 - 3n - 4 =
4(2(2n)) - 3n - 4 =
4(2(2n) - 3n/4 - 1) =
4(2(2n) - 3n - 1 + 9n/4) =
Since the hypothesis is true for n, 2(2n) - 3n - 1 = 9k for some integer, k.
4(9k + 9n/4)
= 36k + 9n
= 9(4k + n) which is divisible by 9.
Therefore, the hypothesis is true for n + 1.
Therefore, by induction, 2(2n) - 3n - 1 is divisible by 9 for all positive integers n.
Hence we can conclude that x is a subset of Y.
This question is part of UPSC exam. View all Class 11 courses
Most Upvoted Answer
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subse...
Understanding the Sets X and Y
Set X is defined as:
- X = {4^n - 3n - 1 : n ∈ N}
Set Y is defined as:
- Y = {9(n - 1) : n ∈ N}
Analyzing Set X
- For n = 1:
X = 4^1 - 3(1) - 1 = 4 - 3 - 1 = 0
- For n = 2:
X = 4^2 - 3(2) - 1 = 16 - 6 - 1 = 9
- For n = 3:
X = 4^3 - 3(3) - 1 = 64 - 9 - 1 = 54
- For n = 4:
X = 4^4 - 3(4) - 1 = 256 - 12 - 1 = 243
Analyzing Set Y
- For n = 1:
Y = 9(1 - 1) = 0
- For n = 2:
Y = 9(2 - 1) = 9
- For n = 3:
Y = 9(3 - 1) = 18
- For n = 4:
Y = 9(4 - 1) = 27
- For n = 5:
Y = 9(5 - 1) = 36
Proving X is a Subset of Y
To prove that X ⊆ Y, we need to show that every element of X is also an element of Y.
- Observation:
The values in set X for n = 1, 2, 3, 4 yield 0, 9, 54, 243.
The values in set Y yield 0, 9, 18, 27, 36, ...
- Induction:
For larger n, the expression 4^n grows faster than the linear term 3n + 1, implying all results from X will remain multiples of 9.
- Conclusion:
Thus, each evaluated element of X corresponds to an element in Y, confirming X is a subset of Y.
Set X is defined as:
- X = {4^n - 3n - 1 : n ∈ N}
Set Y is defined as:
- Y = {9(n - 1) : n ∈ N}
Analyzing Set X
- For n = 1:
X = 4^1 - 3(1) - 1 = 4 - 3 - 1 = 0
- For n = 2:
X = 4^2 - 3(2) - 1 = 16 - 6 - 1 = 9
- For n = 3:
X = 4^3 - 3(3) - 1 = 64 - 9 - 1 = 54
- For n = 4:
X = 4^4 - 3(4) - 1 = 256 - 12 - 1 = 243
Analyzing Set Y
- For n = 1:
Y = 9(1 - 1) = 0
- For n = 2:
Y = 9(2 - 1) = 9
- For n = 3:
Y = 9(3 - 1) = 18
- For n = 4:
Y = 9(4 - 1) = 27
- For n = 5:
Y = 9(5 - 1) = 36
Proving X is a Subset of Y
To prove that X ⊆ Y, we need to show that every element of X is also an element of Y.
- Observation:
The values in set X for n = 1, 2, 3, 4 yield 0, 9, 54, 243.
The values in set Y yield 0, 9, 18, 27, 36, ...
- Induction:
For larger n, the expression 4^n grows faster than the linear term 3n + 1, implying all results from X will remain multiples of 9.
- Conclusion:
Thus, each evaluated element of X corresponds to an element in Y, confirming X is a subset of Y.
Attention Class 11 Students!
To make sure you are not studying endlessly, EduRev has designed Class 11 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 11.
|
|
Explore Courses for Class 11 exam
|
|
Top Courses for Class 11View all
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y
Question Description
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y.
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y.
Solutions for if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y in English & in Hindi are available as part of our courses for Class 11.
Download more important topics, notes, lectures and mock test series for Class 11 Exam by signing up for free.
Here you can find the meaning of if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y defined & explained in the simplest way possible. Besides giving the explanation of
if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y, a detailed solution for if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y has been provided alongside types of if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y theory, EduRev gives you an
ample number of questions to practice if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y tests, examples and also practice Class 11 tests.
|
|
Explore Courses for Class 11 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup