Sequences Can Start at n = 0 Examples

Find a formula for the general term of the sequence

1, 2, 3, 4,...

if the sequence starts at

(a) n = 1

(b) n = 0

(c) n = 194

(a) This is the easy one. The formula is a_n = n.

(b) Make a table so we can see the relationship between n and the term. This time, the table starts with n = 0 instead of n = 1.

The formula for the general term if we start at n = 0 is a_n = n + 1.

(c) Let's make a table so we can see the relationship. Start the table off at n = 194.

If we start this sequence at n = 194, the n^th term is a_n = n – 193.

Find a formula for the general term of the sequence

Answer 1.

If we look at the denominators first we see the sequence

1, 2, 3, 4,...

It makes sense to start this sequence at n = 1. In order for the first term to be positive and the terms to alternate sign, we must multiply by (-1)^{n + 1}.

The general term of the sequence, starting at n = 1, is

Answer 2.

If we look at the signs first we see

1,-1,1,-1,....

This can be described by (-1)ⁿ, starting at n = 0. Then the denominator of the n^th term is (n + 1). The general term, starting at n = 0, is