Question

Question #fb8ba

Answer 1

If you are referring to the sequence formula for the sequence defined above, consider the two different patterns you can see happening, and construct a general term `x_n` from that pattern.

First, if we ignore the signs, we see the sequence is an odd numbers sequence: 1, 3, 5, 7, 9. If we use `n` to refer to the nth term, with `n = 1, 2, 3...`, then we can see that the nth odd number is given by `(2n - 1)`.

What about the signs? Well, the signs are alternating from positive to negative, with the negatives happening when `n` is an even number. Thus, `x_1` is positive, `x_2` is negative, `x_3` is positive, etc. A way to deal with this is to use the expression `(-1)^(n+1)`.

Multiplying these two expressions together gives us a formula for the term `x_n`:

`x_n = (-1)^(n+1)(2n - 1), n = 1,2,3...`

Note:

Some people like to have the first term in a sequence be the `n = 0` term (rather than starting with `n = 1`. In that case, the formula would change slightly:

`x_n = (-1)^n(2n+1), n = 0, 1, 2, ...`