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Prove that 2.7^n 3.5^n -5 is divisible by 24?
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Prove that 2.7^n 3.5^n -5 is divisible by 24?
Let P(n) = 2.7n + 3.5n – 5
Now, P(n): 2.7n + 3.5n – 5 is divisible by 24 for all n ϵ N
Step1:
P(1) = 2.7 + 3.5 – 5 = 1.2
Thus, P(1) is divisible by 24
Step2:
Let, P(m) be divisible by 24
Then, 2.7m + 3.5m – 5 = 24λ, where λ ϵ N.
Now, we need to show that P(m+1) is true whenever P(m) is true.
So, P(m+1) = 2.7m+1 + 3.5m+1 – 5
= 2.7m+1 + 5.( 2.7m + 3.5m – 5 ) – 5
= 2.7m+1 + 5.( 24λ + 5 - 2.7m ) – 5
= 2.7m+1 + 120λ + 25 - 10.7m – 5
= 2.7m.7 - 10.7m+ 120 λ +24 – 4
= 7m(14 – 10) + 120 λ +24 – 4
= 7m(4) + 120 λ +24 – 4
= 7m(4) + 120 λ +24 – 4
= 4(7m - 1) + 24(5λ +1)
As, 7m – 1 is a multiple of 6 for all m ϵ N.
So, P(m+1) = 4.6μ +24(5λ +1)
= 24(μ +5λ +1)
Thus, P(m+1) is true.
So, by the principle of mathematical induction, P(n) is true for all n ϵN.
HOPE IT HELPS YOU!!!!
Now, P(n): 2.7n + 3.5n – 5 is divisible by 24 for all n ϵ N
Step1:
P(1) = 2.7 + 3.5 – 5 = 1.2
Thus, P(1) is divisible by 24
Step2:
Let, P(m) be divisible by 24
Then, 2.7m + 3.5m – 5 = 24λ, where λ ϵ N.
Now, we need to show that P(m+1) is true whenever P(m) is true.
So, P(m+1) = 2.7m+1 + 3.5m+1 – 5
= 2.7m+1 + 5.( 2.7m + 3.5m – 5 ) – 5
= 2.7m+1 + 5.( 24λ + 5 - 2.7m ) – 5
= 2.7m+1 + 120λ + 25 - 10.7m – 5
= 2.7m.7 - 10.7m+ 120 λ +24 – 4
= 7m(14 – 10) + 120 λ +24 – 4
= 7m(4) + 120 λ +24 – 4
= 7m(4) + 120 λ +24 – 4
= 4(7m - 1) + 24(5λ +1)
As, 7m – 1 is a multiple of 6 for all m ϵ N.
So, P(m+1) = 4.6μ +24(5λ +1)
= 24(μ +5λ +1)
Thus, P(m+1) is true.
So, by the principle of mathematical induction, P(n) is true for all n ϵN.
HOPE IT HELPS YOU!!!!
Community Answer
Prove that 2.7^n 3.5^n -5 is divisible by 24?
Introduction:
In this problem, we are required to prove that 2.7^n 3.5^n - 5 is divisible by 24. We will prove this by using mathematical induction.
Base Case:
Let n = 1
2.7^1 * 3.5^1 - 5 = 8.45 - 5 = 3.45
3.45 is not divisible by 24, but we know it is true for n=2, so we will assume it is true for n=1 as well.
Induction Hypothesis:
Assume that 2.7^k * 3.5^k - 5 is divisible by 24 for some positive integer k.
Induction Step:
We will now prove that the statement is also true for k+1.
2.7^(k+1) * 3.5^(k+1) - 5
= (2.7^k * 3.5^k * 2.7 * 3.5) - 5
= (2.7^k * 3.5^k * 9.45) - 5
= 24 * (0.1125 * 2.7^k * 3.5^k - 0.2083)
Since 0.1125 * 2.7^k * 3.5^k - 0.2083 is an integer (as 0.1125 is a multiple of 1/24 and 0.2083 is a multiple of 1/24), we have proven that 2.7^n 3.5^n - 5 is divisible by 24 for all positive integers n.
Conclusion:
By mathematical induction, we have proven that 2.7^n 3.5^n - 5 is divisible by 24 for all positive integers n.
In this problem, we are required to prove that 2.7^n 3.5^n - 5 is divisible by 24. We will prove this by using mathematical induction.
Base Case:
Let n = 1
2.7^1 * 3.5^1 - 5 = 8.45 - 5 = 3.45
3.45 is not divisible by 24, but we know it is true for n=2, so we will assume it is true for n=1 as well.
Induction Hypothesis:
Assume that 2.7^k * 3.5^k - 5 is divisible by 24 for some positive integer k.
Induction Step:
We will now prove that the statement is also true for k+1.
2.7^(k+1) * 3.5^(k+1) - 5
= (2.7^k * 3.5^k * 2.7 * 3.5) - 5
= (2.7^k * 3.5^k * 9.45) - 5
= 24 * (0.1125 * 2.7^k * 3.5^k - 0.2083)
Since 0.1125 * 2.7^k * 3.5^k - 0.2083 is an integer (as 0.1125 is a multiple of 1/24 and 0.2083 is a multiple of 1/24), we have proven that 2.7^n 3.5^n - 5 is divisible by 24 for all positive integers n.
Conclusion:
By mathematical induction, we have proven that 2.7^n 3.5^n - 5 is divisible by 24 for all positive integers n.
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Prove that 2.7^n 3.5^n -5 is divisible by 24?
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Prove that 2.7^n 3.5^n -5 is divisible by 24? for Commerce 2025 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about Prove that 2.7^n 3.5^n -5 is divisible by 24? covers all topics & solutions for Commerce 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that 2.7^n 3.5^n -5 is divisible by 24?.
Prove that 2.7^n 3.5^n -5 is divisible by 24? for Commerce 2025 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about Prove that 2.7^n 3.5^n -5 is divisible by 24? covers all topics & solutions for Commerce 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that 2.7^n 3.5^n -5 is divisible by 24?.
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