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if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y
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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.
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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.
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if x=(4 ^n - 3n - 1 :n €N } and Y={9(n-1):N€N},prove that X is a subset of Y
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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.
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