Hello, shanaz!

This is an exercise in Imagination.

There are no formulas for it.

Mrs.James put this sequence on the board: 1, 2, 3, . . .

She asked her class to show how it continued.

Penny wrote down: 1, 2, 3, 4, 5, 6, ...

Blair wrote down: 1, 2, 3, 5, 8,13, ...

a. Explain how Penny and Blair got their sequences.

Penny guessed that each term is one more than the preceding term.

Blair guessed that each term is sum of the two preceding terms.

b. How might the sequence 1, 2, 4, ... continue?

Give more than one way.

One possibility is that each term is twice the preceding term.

. . So the sequence is: .

Another is that the differences increase by one.

That is, the sequence could be generated like this:

So the sequence might be: .

C. How might the sequence 3, 8, 13, ... continue?

Give more than one way.

The obvious rule: each term is 5 more than the preceding term.

The sequence continues: .

Finding a second reason requires some algebraic theory.

Here's one solution (with no explanation):

. . The generating function could be: .

For , we have: .