What is the number directly below 99 when the triangular array below is continued?

1
2 3
4 5 6
7 8 9 10
11 12 13 14 15

114

To find the number directly below 99 in the triangular array, we need to determine the row and position of 99 in the array.

Let's observe the pattern in the numbers within the array. Starting from the top, each row contains consecutive numbers, incrementing by 1.

1
2 3
4 5 6
7 8 9 10
11 12 13 14 15

By counting the numbers, we can see that 99 is located in the 14th row. To calculate the position of 99 within this row, we need to find the total number of elements in the previous rows.

In total, there are (1+2+3+...+13) + 14 elements in the triangular array before row 14. This sum can be calculated using the formula for the sum of an arithmetic series:

Sum = (n/2) * (first term + last term)
= (13/2) * (1 + 13)
= 91 * 14
= 1274

So, there are 1274 elements before row 14. Now, we can calculate the position of 99 within row 14.

The first number in row 14 is 1274 + 1 = 1275. Since each number in the row increases by 1, we can calculate the position of 99 by subtracting the first number in row 14 from 99 and adding 1.

Position of 99 = 99 - 1275 + 1
= -1175 + 1
= -1174

Since the position is negative, there is no number directly below 99 when the triangular array is continued.