(a)

Observe the figure:

The first figure has dot, dots, dots and dots.

The are square of the first four natural numbers.

Therefore the next figure consist of dots.

 (b)

There is dot in the first figure, dots in the second figure, dots in the third figure and dots in the fourth figure.

.

.

.

.

Therefore, the sequece is . 

(c)

The sequece is .

Each term in the sequence can be found by subtracting from .

Explicit formula for the sequence is .

(d)

The sequece is .

Rewrite sequence as

Expression for the number of dots in the figure in the original sequence is .

(e)

Let be statement .

The statement is true for , since .

Assume that statement is true for .

Final statement is exactly .

Thus, is true.

Since   is true for and implies .

is true for .

Therefore by the mathematical induction is true for all

positive integers .

(a) The figure is

(b) The sequece is .

(c) Explicit formula for the sequence is .

(d) Number of dots in the figure in the original sequence is .

(e) .