The next term in the series 0, 1, 9, 36, 100, ______ is:
225
To identify the next term in the given series: 0, 1, 9, 36, 100, ______, we need to determine the pattern or rule governing the series.
Let's analyze the given terms:
- 0: This is the square of 0. \(0^2 = 0\)
- 1: This is the square of 1. \(1^2 = 1\)
- 9: This is the square of 3. \(3^2 = 9\)
- 36: This is the square of 6. \(6^2 = 36\)
- 100: This is the square of 10. \(10^2 = 100\)
The series appears to be the square of consecutive triangular numbers. Triangular numbers are those which can form an equilateral triangle. The sequence of triangular numbers is: 0, 1, 3, 6, 10, 15, ...
Let's calculate the next triangular number after 10:
- The triangular number after \(10\) is \(15\).
Therefore, the next term in the series, which is the square of this triangular number, is:
- \(15^2 = 225\)
Thus, the next term in the series is 225. Hence, the correct answer is 225.
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