Question

the ancient greeks labeled certain numbers as triangular numbers. the numbers 1, 3, 6, 10, 15, 21, and so on are triangular numbers. complete parts a) through c) below.
(images of triangular dot patterns labeled 1, 3, 6, 10, 15, 21)
...
c. add the last two triangular numbers together to get the next one.
d. double the previous triangular number and then subtract half of the previous from that
c) is 72 a triangular number? explain how you determined your answer. choose the correct answer below.
a. no, because it is an even number
b. no, because it falls between two triangular numbers, 66 and 78.
c. yes, because you can make a triangular shape with 72 dots.
d. yes, because it is 9 squared.

Explanation:

Step1: Recall triangular number formula

The nth triangular number is given by $T_n = \frac{n(n+1)}{2}$

Step2: Set formula equal to 72

Set $\frac{n(n+1)}{2} = 72$, multiply both sides by 2:
$n(n+1) = 144$

Step3: Rearrange to quadratic equation

Expand and rearrange: $n^2 + n - 144 = 0$

Step4: Solve quadratic equation

Use quadratic formula $n = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ where $a=1, b=1, c=-144$:
$n = \frac{-1 \pm \sqrt{1 + 576}}{2} = \frac{-1 \pm \sqrt{577}}{2}$
$\sqrt{577} \approx 24.02$, so positive solution is $\frac{-1 + 24.02}{2} \approx 11.51$

Step5: Check nearby triangular numbers

Calculate $T_{11} = \frac{11\times12}{2}=66$, $T_{12}=\frac{12\times13}{2}=78$
72 falls between 66 and 78, with no integer n giving $T_n=72$.

Answer:

B. No, because it falls between two triangular numbers, 66 and 78