Sequences Can Start at n = 0 - At A Glance

Remember the Count from Sesame Street? Now a little bit older and with a few gray hairs, he needs a good pick-me-up in the morning, "One. One large coffee. Ha.Ha.Ha." Why would we begin counting sequences from zero? The Count never started counting from 0.

Sometimes it's easier. Computer programmers start from 0 all the time, and a sequence can start from the zeroth term, n = 0, too. It doesn't matter. We have the freedom to start from whatever number n we want. We can be rebels and begin from n = 194.

We need to keep mind, though, that the formula for the general term of a sequence will vary depending on the starting value of n. While we can start a sequence at any value of n we like, n = 0 and n = 1 are the most common starting values. They are the easiest to work with. And they are the two neatest numbers we have.

Here are some variations on the common sequences that show up when we start a sequence at n = 0:

• The sequence of whole numbers numbers a_n = n:
  0, 1, 2, 3, 4,...
    
• The sequence of even numbers a_n = 2n:
  0, 2, 4, 6, 8,...
    
• The sequence of odd numbers a_n = 2n + 1:
  1, 3, 5, 7,...
    
• The sequence of powers of 2 where a_n = 2ⁿ:
  1, 2, 4, 8, 16, 32,...

Formulas for alternating sequences of ±1 also get affected by the starting value of n. When we started at n = 1, the sequence

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

was given by

a_n = (-1)ⁿ.

If instead we start at n = 0, this sequence is given by

a_n = (-1)^{n + 1}.

To make sure you have the correct exponent on your (-1) factor, evaluate the first term of the sequence and make sure it has the correct sign.

If you're asked to find the formula for the general term of a sequence and not given the starting value of n, you should choose n = 0 or n = 1. It can seem a bit like choosing between plain or peanut M&Ms. If you are in the mood for a chocolatey start and a peanutty finish, start from n = 0. And sometimes, one starting value may seem more convenient than the other.

Sample Problem

Find a formula for the general term of the sequence

Answer 1.

If we look at the numerators first, we see the sequence

1, 2, 4, 8,...

These are the powers of 2 starting at n = 0. Take n = 0 as the starting point and look at the denominators:

The denominator is given by (n + 1)!. The sequence is given by

, starting at n = 0.

Answer 2.

If we look at the denominators first, we see the sequence

1!, 2!, 3!, 4!, ...

These are the factorials, starting at n = 1. Take n = 1 as the starting point and look at the numerators:

The numerator is 2^{n – 1}. The sequence is given by

, starting at n = 1.

In the example above, we got two possible formulas depending on the starting value of n:

or

In the first formula, the exponent (n) is simple and the factorial ((n + 1)!) is complicated. In the second formula, the exponent (n – 1) is complicated and the factorial (n!) is simple. In the end, it doesn't matter which formula you pick. You have the power--pun intended--to begin wherever you like.

Sometimes one formula and starting value will be preferable over the other. To write the even numbers

0, 2, 4, 6, 8,...

we want to start with n = 0 so the general term can be

a_n = 2n. If we started with n = 1 the general term would be

a_n = 2n – 2, which is more complicated than necessary.

In cases like this, use the simple answer. Picking n so you have a simpler formula will save you time. Maybe you can use that extra time to stick your younger brother's favorite shoes to the ceiling with super glue.