To determine if 72 is a triangular number, we need to check if there exists an integer [tex]\( n \)[/tex] such that the [tex]\( n \)[/tex]-th triangular number [tex]\( T_n \)[/tex] equals 72. The formula for calculating the [tex]\( n \)[/tex]-th triangular number is:
[tex]\[ T_n = \frac{n(n + 1)}{2} \][/tex]
Given the number 72, we need to solve the equation:
[tex]\[ \frac{n(n + 1)}{2} = 72 \][/tex]
Multiplying both sides of the equation by 2 to clear the fraction gives:
[tex]\[ n(n + 1) = 144 \][/tex]
This is a quadratic equation in terms of [tex]\( n \)[/tex]:
[tex]\[ n^2 + n - 144 = 0 \][/tex]
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In our equation [tex]\( n^2 + n - 144 = 0 \)[/tex], [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -144 \)[/tex]. Plugging these into the quadratic formula, we get:
[tex]\[ n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \][/tex]
[tex]\[ n = \frac{-1 \pm \sqrt{1 + 576}}{2} \][/tex]
[tex]\[ n = \frac{-1 \pm \sqrt{577}}{2} \][/tex]
Calculating the square root of 577 gives approximately 24.02. Therefore,
[tex]\[ n = \frac{-1 \pm 24.02}{2} \][/tex]
This yields two solutions:
[tex]\[ n = \frac{23.02}{2} \approx 11.51 \][/tex]
[tex]\[ n = \frac{-25.02}{2} \approx -12.51 \][/tex]
Since [tex]\( n \)[/tex] must be a positive integer, the negative solution is irrelevant. The positive solution [tex]\( n \approx 11.51 \)[/tex] is not an integer. Hence, 72 is not a triangular number.
To confirm this, we can observe that the triangular numbers around 72 are:
- For [tex]\( n = 11 \)[/tex]:
[tex]\[ T_{11} = \frac{11 \cdot 12}{2} = 66 \][/tex]
- For [tex]\( n = 12 \)[/tex]:
[tex]\[ T_{12} = \frac{12 \cdot 13}{2} = 78 \][/tex]
Therefore, 72 falls between two triangular numbers, 66 and 78.
Hence, the correct answer is:
OC. No, because it falls between two triangular numbers, 66 and 78.
