Question

How many triangular numbers are there between 11 and 39?

Answer 1

Triangular numbers are given by the formula:

\[
T_n = \frac{n(n+1)}{2}
\]

where \( n \) is a positive integer. We want to find the values of \( n \) for which \( 11 < T_n < 39 \).

1. **Find the triangular numbers greater than 11**:
We'll calculate \( T_n \) for \( n = 1, 2, 3, \ldots \) until \( T_n \) exceeds 39.

- For \( n = 1\):
\[
T_1 = \frac{1(1+1)}{2} = 1
\]
- For \( n = 2\):
\[
T_2 = \frac{2(2+1)}{2} = 3
\]
- For \( n = 3\):
\[
T_3 = \frac{3(3+1)}{2} = 6
\]
- For \( n = 4\):
\[
T_4 = \frac{4(4+1)}{2} = 10
\]
- For \( n = 5\):
\[
T_5 = \frac{5(5+1)}{2} = 15
\]
- For \( n = 6\):
\[
T_6 = \frac{6(6+1)}{2} = 21
\]
- For \( n = 7\):
\[
T_7 = \frac{7(7+1)}{2} = 28
\]
- For \( n = 8\):
\[
T_8 = \frac{8(8+1)}{2} = 36
\]
- For \( n = 9\):
\[
T_9 = \frac{9(9+1)}{2} = 45
\]

Since \( T_9 = 45 \) exceeds 39, we stop here.

2. **Identify the triangular numbers between 11 and 39**:
From our calculations:

- The triangular numbers we found are: \( 1, 3, 6, 10, 15, 21, 28, 36, 45 \)
- The ones that are between 11 and 39 are: \( 15, 21, 28, \) and \( 36 \).

Thus, the triangular numbers between 11 and 39 are \( 15, 21, 28, \) and \( 36 \).

3. **Count the numbers**:
There are **4 triangular numbers** between 11 and 39.

\[
\boxed{4}
\]