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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
(2019)
(2019)
- a)20
- b)15
- c)10
- d)25
Correct answer is option 'C'. 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)...
Case 1 - if m = odd then, (m + 1) = even
⇒ f(m) = m + 3 and f(m + 1) = m(m + 1)
⇒ 8 × (m + 2)(m + 1) - (m + 3) = 2
⇒ 8m2 + 23m - 11 = 0
⇒ m = [-23 +-√(23)2 - 4 × 11 × 8]/18
⇒ Here the value of m is not integer so this case is wrong
⇒ Case 2 - if m = even and m + 1 = odd
⇒ f(m + 1) = m + 3 and f(m) = m(m + 1)
⇒ 8(m + 1 + 3) - m(m + 1) = 2
⇒ m2 - 7m - 30 = 0
⇒ (m + 3)(m - 10) = 0
∴ As m = even so m = 10
Most Upvoted Answer
For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n)...
Case 1: m is even
Given, 8f(m + 1) – f(m) = 2
⇒ 8(m + 1 + 3) – m(m + 1) = 2
⇒ 8m + 32 – m2 – m = 2
⇒ m2 – 7m – 30 = 0
⇒ (m – 10)(m + 3) = 0
⇒ m = 10 or –3
As m is a positive integer, therefore m = 10.
Case 2: If m is odd, then
8f(m + 1) – f(m) = 2
⇒ 8(m + 1)(m + 2) – (m + 3) = 2
⇒ 8(m2 + 3m + 2) – m – 3 = 2
⇒ 8m2 + 24m + 16 – m – 3 = 2
⇒ 8m2 + 23m + 11 = 0,
Which is not possible and hence no solution.
Given, 8f(m + 1) – f(m) = 2
⇒ 8(m + 1 + 3) – m(m + 1) = 2
⇒ 8m + 32 – m2 – m = 2
⇒ m2 – 7m – 30 = 0
⇒ (m – 10)(m + 3) = 0
⇒ m = 10 or –3
As m is a positive integer, therefore m = 10.
Case 2: If m is odd, then
8f(m + 1) – f(m) = 2
⇒ 8(m + 1)(m + 2) – (m + 3) = 2
⇒ 8(m2 + 3m + 2) – m – 3 = 2
⇒ 8m2 + 24m + 16 – m – 3 = 2
⇒ 8m2 + 23m + 11 = 0,
Which is not possible and hence no solution.
Community Answer
For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n)...
Understanding the Function f(n)
The function f(n) is defined differently based on whether n is even or odd:
- If n is even: f(n) = n(n + 1)
- If n is odd: f(n) = n + 3
Setting Up the Equation
Given the equation:
8f(m + 1) - f(m) = 2
We need to evaluate f(m) and f(m + 1) based on whether m is even or odd.
Case 1: m is even
- f(m) = m(m + 1)
- m + 1 is odd, thus f(m + 1) = (m + 1) + 3 = m + 4
Substituting into the equation:
8(m + 4) - m(m + 1) = 2
Simplifying:
8m + 32 - m^2 - m = 2
Rearranging gives:
-m^2 + 7m + 30 = 0
Using the quadratic formula or factoring, you find:
(m - 10)(m + 3) = 0
This gives m = 10 (since m must be positive).
Case 2: m is odd
- f(m) = m + 3
- m + 1 is even, thus f(m + 1) = (m + 1)(m + 2)
Substituting into the equation:
8(m + 1)(m + 2) - (m + 3) = 2
This simplifies to a more complex polynomial, but we focus on the case where m is even since we've already reached a solution.
Conclusion
The only valid solution for m is 10, which aligns with option C.
Thus, m equals 10.
The function f(n) is defined differently based on whether n is even or odd:
- If n is even: f(n) = n(n + 1)
- If n is odd: f(n) = n + 3
Setting Up the Equation
Given the equation:
8f(m + 1) - f(m) = 2
We need to evaluate f(m) and f(m + 1) based on whether m is even or odd.
Case 1: m is even
- f(m) = m(m + 1)
- m + 1 is odd, thus f(m + 1) = (m + 1) + 3 = m + 4
Substituting into the equation:
8(m + 4) - m(m + 1) = 2
Simplifying:
8m + 32 - m^2 - m = 2
Rearranging gives:
-m^2 + 7m + 30 = 0
Using the quadratic formula or factoring, you find:
(m - 10)(m + 3) = 0
This gives m = 10 (since m must be positive).
Case 2: m is odd
- f(m) = m + 3
- m + 1 is even, thus f(m + 1) = (m + 1)(m + 2)
Substituting into the equation:
8(m + 1)(m + 2) - (m + 3) = 2
This simplifies to a more complex polynomial, but we focus on the case where m is even since we've already reached a solution.
Conclusion
The only valid solution for m is 10, which aligns with option C.
Thus, m equals 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 equals(2019)a)20b)15c)10d)25Correct answer is option 'C'. Can you explain this answer?
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