The first four triangular numbers $1,3,6$ and 10 are illustrated in the diagram. What is the tenth triangular number?
(A) 55
(D) 78
(B) 45
(C) 66
(E) 50
![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-138.jpg?height=204&width=624&top_left_y=1752&top_left_x=1192)
The first four triangular numbers $1,3,6$, and 10 are illustrated in the diagram. What is the tenth triangular number?
(A) 55
(D) 78
(B) 45
(C) 66
(E) 50

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-254.jpg?height=234&width=737&top_left_y=1737&top_left_x=1190)

## Solution

From our diagram, it can be seen that the fifth triangular number is found by adding a row with five dots. This number is thus, $1+2+3+4+5=\frac{5 \times 6}{2}=15$. The sixth triangular number is $1+2+3+4+5+6=\frac{6 \times 7}{2}=21$. If we follow this to its conclusion, the $n$th triangular number is
$1+2+3+\ldots+n=\frac{(n)(n+1)}{2}$. The tenth triangular number will be $\frac{(10)(11)}{2}=55$.

ANSWER: (A)