If \(x_m +1\) and \(x_m=x_{m+1}+(m+1)\) for every positive integer \(m\), then \(x_{100 }\) equals
- -5151
- -5150
- -5051
- -5050
The Correct Option is D
Approach Solution - 1
Given:
A recursive relation: \[ x_{m+1} = x_m - (m + 1) \] with base value: \[ x_1 = -1 \]
Step-by-step Calculation:
Using the recurrence:
- \[ x_2 = x_1 - 2 = -1 - 2 = -3 \]
- \[ x_3 = x_2 - 3 = -3 - 3 = -6 \]
- \[ x_4 = x_3 - 4 = -6 - 4 = -10 \]
- and so on...
From the pattern, we see: \[ x_n = - (1 + 2 + 3 + \dots + n) = -\frac{n(n+1)}{2} \]
Final Calculation:
To find: \[ x_{100} = -\frac{100 \times 101}{2} = -5050 \]
Correct Answer:
\(\boxed{-5050}\)
Approach Solution -2
Given:
\[ x_1 = -1, \quad \text{and} \quad x_m = x_{m+1} + (m+1) \] Rearranging the recursive formula: \[ x_{m+1} = x_m - (m+1) \]
Step-by-Step Calculation:
Let’s compute the first few values:
- \( x_1 = -1 \) (Given)
- \( x_2 = x_1 - 2 = -1 - 2 = -3 \)
- \( x_3 = x_2 - 3 = -3 - 3 = -6 \)
- \( x_4 = x_3 - 4 = -6 - 4 = -10 \)
- \( x_5 = x_4 - 5 = -10 - 5 = -15 \)
- \( x_6 = x_5 - 6 = -15 - 6 = -21 \)
- ... and so on ...
Clearly, the pattern becomes: \[ x_n = - (1 + 2 + 3 + \dots + n) \] Which is the negative of the sum of the first \( n \) natural numbers.
General Formula:
\[ x_n = -\frac{n(n+1)}{2} \]
Find \( x_{100} \):
\[ x_{100} = -\frac{100(101)}{2} = -50 \times 101 = \boxed{-5050} \]
Final Answer:
The correct option is (D): \[ \boxed{-5050} \]
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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.
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