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
- 13
- 12
- 11
- 10
The Correct Option is D
Solution and Explanation
Solve the equation:
\(8f(m + 1) - f(m) = 2\)
where \(f(m) = m(m + 1)\) when \(m\) is even, and \(f(m) = m + 3\) when \(m\) is odd.
Case 1: When \(m\) is even
Since \(m\) is even, we use: \(f(m) = m(m + 1)\)
and \(m + 1\) is odd, so: \(f(m + 1) = (m + 1) + 3 = m + 4\)
Substitute into the original equation:
Set equal to 2:
Solve the quadratic:
\[(m - 10)(m + 3) = 0 \Rightarrow m = 10 \text{ or } m = -3\]Only the positive integer solution is: \(\boxed{m = 10}\)
Case 2: When \(m\) is odd
Then \(f(m) = m + 3\) and \(f(m + 1) = (m + 1)(m + 2)\) since \(m + 1\) is even.
Substituting into the equation:
\[8f(m + 1) - f(m) = 8(m + 1)(m + 2) - (m + 3)\]
Set equal to 2:
Set equal to 2:
\[8m^2 + 23m + 13 = 2 \Rightarrow 8m^2 + 23m + 11 = 0\]The discriminant is:
\[D = 23^2 - 4 \cdot 8 \cdot 11 = 529 - 352 = 177\]Since the discriminant is not a perfect square, there is no integer solution.
Conclusion:
The only valid solution is when \(m = 10\).
Final Answer: \(\boxed{m = 10}\)
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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
- Para Summary
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