Question
Triangular numbers can be represented with equilateral triangles formed by dots. The first five triangular numbers are 1, 3, 6, 10, and 15. Is there a direct variation between a triangular number and its position in the sequence? Explain your reasoning.
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Answers
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GPT 3.5
Yes, there is a direct variation between a triangular number and its position in the sequence.
The nth triangular number can be calculated using the formula Tn = n(n+1) / 2, where n represents the position in the sequence.
For example, the 4th triangular number can be calculated by plugging in n=4 into the formula: T4 = 4(4+1) / 2 = 10.
As seen in the above example, the triangular number increases as the position in the sequence increases by a constant rate according to the formula. Therefore, there is a direct variation between a triangular number and its position in the sequence.
The nth triangular number can be calculated using the formula Tn = n(n+1) / 2, where n represents the position in the sequence.
For example, the 4th triangular number can be calculated by plugging in n=4 into the formula: T4 = 4(4+1) / 2 = 10.
As seen in the above example, the triangular number increases as the position in the sequence increases by a constant rate according to the formula. Therefore, there is a direct variation between a triangular number and its position in the sequence.
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