The first three triangular numbers are and .

(a)

From the pattern of dots, we can see that each triangle contains one greater number of dots than the previous row.

row: .

row: .

row: .

row: .

row: .

(b)

The sequence of numbers shows a pattern of second common difference.

This means there exists a quadratic expression of the form  to model this sequence.

Substitute and values in .

.

.

.

Solve the system of linear equations.

Subtract equation from .

Subtract equation from .

Subtract equation from .

.

Substitute in equation .

.

Substitute and in equation .

.

Substitute , and in .

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The quadratic model equation is .

(c)

The conjecture is .

Let be the statement that .

Verify that is true for .

Substitute in .

is true for .

Assume that is true for .

Substitute in .

.

is true for positive integer .

Show that must be true.

Add to each side.

.

The final statement is exactly , so is true.

Because is true for and implies , is true for and so on.

That is, by the principle of mathematical induction,

is true for all positive integers .

(a)

row: .

row: .

row: .

row: .

row: .

(b) term of the sequence is .

(c) By the principle of mathematical induction,

is true for all positive integers .