Determining the general term

The general term is given by 3r+1+(-1)^r+1

So by substituting r=0,1,2,3.... I get a sequence like this: 0, 5, 6, 11, 12, 17, 18.....

It seems to form some pattern. So i wonder, can i deduce the general term with only the sequence? How?

Please kindly elaborate more if possible because i really keen to learn how.(Rofl)

Re: Determining the general term

Quote:

Originally Posted by

**MichaelLight** The general term is given by 3r+1+(-1)^r+1

So by substituting r=0,1,2,3.... I get a sequence like this: 0, 5, 6, 11, 12, 17, 18.....

It seems to form some pattern. So i wonder, can i deduce the general term with only the sequence? How?

Please kindly elaborate more if possible because i really keen to learn how.(Rofl)

The sequence can be described as follows:

Consider the rth term (r starts from 0).

If r is even, .

If r is odd, .

So, we need to subtract 1 from 3r+1 if r is even and add 1 to 3r+1 if r is odd. An easy way to do this would be to add to 3r+1.

Thus, we find that the general term is given by .

Re: Determining the general term

Hello, MichaelLight!

When given an increasing sequence of terms, I usually take the difference

. . of consecutive terms. Then the differences of the differences, and so on.

We hope to find that the differences are constant.

. . This indicates that the generating function is an degree polynomial.

So we have:

. .

But we see that the 1st difference has *alternating* terms.

This indicates that the sequence is composed of two subsequences,

. . one for "even" terms, one for "odd" terms.

We have:

. .

We see that, for even

. . . . . . . . . . for odd

And from here, we can consult Alex's excellent explanation.