For Exercises 32 and 33, use the matrix at the right.

$\left[\begin{array}{cccccc}1& 3& 6& 10& 15& \mathrm{...}\\ 2& 5& 9& 14& 20& \mathrm{...}\\ 4& 8& 13& 19& 26& \mathrm{...}\\ 7& 12& 18& 25& 33& \mathrm{...}\\ 11& 17& 24& 32& 41& \mathrm{...}\\ 16& 23& 31& 40& 50& \mathrm{...}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right]$

Study the pattern of numbers. Complete the matrix for column 6 and row 7.

In the given matrix it can be noticed that in the first row entries are: 1, 3, 6, 10 and 15.

Therefore, it can be noticed that the difference between the entries are: 2, 3, 4 and 5.

Therefore, similarly the sixth entry will be such that the difference between the sixth entry and fifth entry should be 6.

Therefore, the sixth entry is 21.

In the given matrix it can be noticed that in the second row entries are: 2, 5, 9, 14 and 20.

Therefore, it can be noticed that the difference between the entries are: 3, 4, 5 and 6.

Therefore, similarly the sixth entry will be such that the difference between the sixth entry and fifth entry should be 7.

Therefore, the sixth entry is 27.

In the given matrix it can be noticed that in the third row entries are: 4, 8, 13, 19 and 26.

Therefore, it can be noticed that the difference between the entries are: 4, 5, 6 and 7.

Therefore, similarly the sixth entry will be such that the difference between the sixth entry and fifth entry should be 8.

Therefore, the sixth entry is 34.

In the given matrix it can be noticed that in the fourth row entries are: 7, 12, 18, 25 and 33.

Therefore, it can be noticed that the difference between the entries are: 5, 6, 7 and 8.

Therefore, similarly the sixth entry will be such that the difference between the sixth entry and fifth entry should be 9.

Therefore, the sixth entry is 42.

In the given matrix it can be noticed that in the fifth row entries are: 11, 17, 24, 32 and 41.

Therefore, it can be noticed that the difference between the entries are: 6, 7, 8 and 9.

Therefore, similarly the sixth entry will be such that the difference between the sixth entry and fifth entry should be 10.

Therefore, the sixth entry is 51.

In the given matrix it can be noticed that in the sixth row entries are: 16, 23, 31, 40 and 50.

Therefore, it can be noticed that the difference between the entries are: 7, 8, 9 and 10.

Therefore, similarly the sixth entry will be such that the difference between the sixth entry and fifth entry should be 11.

Therefore, the sixth entry is 61.

In the given matrix it can be noticed that in the first column entries are: 1, 2, 4, 7, 11 and 16.

Therefore, it can be noticed that the difference between the entries are: 1, 2, 3, 4 and 5.

Therefore, similarly the seventh entry will be such that the difference between the seventh entry and sixth entry should be 6.

Therefore, the seventh entry is 22.

We can do similar steps for all steps.

In the sixth row differences between the entries were 7, 8, 9, 10 and 11.

Therefore, by continuing the same pattern till now the seventh row will have difference between the entries as 8, 9, 10, 11 and 12.

Therefore, the entries in the seventh row are: 22, 30, 39, 49, 60 and 72.

The complete matrix is:

$\left[\begin{array}{cccccc}1& 3& 6& 10& 15& 21\\ 2& 5& 9& 14& 20& 27\\ 4& 8& 13& 19& 26& 34\\ 7& 12& 18& 25& 33& 42\\ 11& 17& 24& 32& 41& 51\\ 16& 23& 31& 40& 50& 61\\ 22& 30& 39& 49& 60& 72\end{array}\right]$