Enter a number to check if it's a square triangular number, or input a range to generate all square triangular numbers within that range.
A square triangular number is a natural number that is both a triangular number and a perfect square.
Triangular Numbers: These numbers are formed by summing consecutive natural numbers, defined by the formula: \( T_n = \frac{n(n + 1)}{2} \) where \( T_n \) is the \( n \)-th triangular number.
Perfect Squares: These are numbers that can be expressed as the square of an integer, \( n^2 \).
Some common square triangular numbers include: 1, 36, 1225, 41616, 1413721, 48024900, 1631432881.
Solution:
1. Check if 36 is a triangular number:
\( 36 = \frac{n(n + 1)}{2} \)
Solving the equation, \( n = 8 \).
2. Check if 36 is a perfect square:
\( \sqrt{36} = 6 \)
Result:
36 is a square triangular number.
Solution:
1. Check if 144 is a triangular number:
\( 144 = \frac{n(n + 1)}{2} \)
olving the equation, \( n \approx 16.47 \).
Since \( n \) is not an integer, 144 is not a triangular number. So, 144 is not a square triangular number.
Solution:
1. Check if 1225 is a triangular number:
\( 1225 = \frac{n(n + 1)}{2} \)
Solving the equation, \( n = 49 \).
2. Check if 1225 is a perfect square:
\( \sqrt{1225} = 35 \)
Result:
1225 is a square triangular number.