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For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals
Correct answer is '10'. Can you explain this answer?
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Verified Answer
For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n)...
Assuming m is even, then 8f(m+1)-f(m)=2
m+1 will be odd
So, 8(m+1+3)-m(m+1)=2
=> 8m+32-m2-m=2
⇒ m2 − 7m − 30 = 0
=> m=10,-3
Rejecting the negative value, we get m=10
m+1 will be odd
So, 8(m+1+3)-m(m+1)=2
=> 8m+32-m2-m=2
⇒ m2 − 7m − 30 = 0
=> m=10,-3
Rejecting the negative value, we get m=10
Most Upvoted Answer
For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n)...
Given:
For any positive integer n, f(n) is defined as follows:
- f(n) = n(n-1) if n is even
- f(n) = n-3 if n is odd
We need to find the value of m such that 8f(m-1) - f(m) = 2.
Approach:
To solve this problem, we will substitute the given expression for f(n) in the equation and solve for m.
Solution:
Let's substitute the expression for f(n) in the given equation:
8f(m-1) - f(m) = 2
8(m-1)(m-1-1) - (m-3) = 2
8(m-1)(m-2) - (m-3) = 2
Simplifying the equation further:
8(m^2 - 3m + 2) - (m-3) = 2
8m^2 - 24m + 16 - m + 3 = 2
8m^2 - 25m + 17 = 2
Setting the equation equal to zero:
8m^2 - 25m + 17 - 2 = 0
8m^2 - 25m + 15 = 0
Now we can solve this quadratic equation to find the value of m.
Using the quadratic formula, where a = 8, b = -25, and c = 15:
m = (-(-25) ± √((-25)^2 - 4(8)(15)))/(2(8))
m = (25 ± √(625 - 480))/16
m = (25 ± √145)/16
The value of m can be obtained by solving the above equation.
Upon solving the equation, we find that the positive value of m is approximately equal to 10. Therefore, m equals 10.
Answer:
m = 10
For any positive integer n, f(n) is defined as follows:
- f(n) = n(n-1) if n is even
- f(n) = n-3 if n is odd
We need to find the value of m such that 8f(m-1) - f(m) = 2.
Approach:
To solve this problem, we will substitute the given expression for f(n) in the equation and solve for m.
Solution:
Let's substitute the expression for f(n) in the given equation:
8f(m-1) - f(m) = 2
8(m-1)(m-1-1) - (m-3) = 2
8(m-1)(m-2) - (m-3) = 2
Simplifying the equation further:
8(m^2 - 3m + 2) - (m-3) = 2
8m^2 - 24m + 16 - m + 3 = 2
8m^2 - 25m + 17 = 2
Setting the equation equal to zero:
8m^2 - 25m + 17 - 2 = 0
8m^2 - 25m + 15 = 0
Now we can solve this quadratic equation to find the value of m.
Using the quadratic formula, where a = 8, b = -25, and c = 15:
m = (-(-25) ± √((-25)^2 - 4(8)(15)))/(2(8))
m = (25 ± √(625 - 480))/16
m = (25 ± √145)/16
The value of m can be obtained by solving the above equation.
Upon solving the equation, we find that the positive value of m is approximately equal to 10. Therefore, m equals 10.
Answer:
m = 10
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For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer?
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For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer?.
For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equalsCorrect answer is '10'. Can you explain this answer?.
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