Find a formula for

1/2, -1/4, 1/6, -1/8, 1/10...

So I know that the sequence is just adding 2 to the bottom but i don't know how to write it with a formula including n's..im stuck

Well, I don't want to leave you feeling stuck like a pickle in a jar! Let's try to unravel this sequence, shall we?

From what you've observed, it seems that the pattern of the denominators is simply increasing by 2 each time. Let's see if we can make sense of it by introducing some variables!

Let's denote the nth term of the sequence as An. We can express the pattern as:
An = (-1)^(n+1)/(2n)

So, when n is 1, the first term will be (-1)^(1+1)/(2*1) = 1/2. When n is 2, the second term will be (-1)^(2+1)/(2*2) = -1/4. And so on.

Keep in mind though that n starts at 1 (since the sequence doesn't have a zeroth term), and the exponent of (-1) is (n+1) because it starts with a positive term.

Now that we've got it all figured out, I hope you don't feel like a fish out of water anymore!

To find a formula for the given sequence, we observe that the numerator alternates between 1 and -1, while the denominator increments by 2 for each term.

Let's break down the sequence into two parts:
1/2, 1/6, 1/10... (odd terms with numerator 1)
and
-1/4, -1/8, -1/12,... (even terms with numerator -1).

For the odd terms, we notice that the denominator can be expressed as (2n), where n represents the position of the term in the sequence. In other words, for the first term, n = 1, for the second term, n = 2, and so on.

For the even terms, the denominator can be expressed as -(2n-2), where n represents the position of the term in the sequence. In other words, for the first term, n = 1, for the second term, n = 2, and so on.

To simplify this, we can express the formula for the entire sequence as follows:

For odd terms:
(-1)^(n+1) / (2n)

For even terms:
(-1)^n / (2n-2)

Combining these two formulas, we get the following formula for the sequence:
(-1)^((n+1)+floor((n+1)/2)) / (2n-floor((n+1)/2))

Note: "floor" refers to the floor function, which rounds down to the nearest integer.

This formula will generate the terms of the given sequence.

This is a geometric sequence where

a=1/2 and r = -1/2

t(n) = ar^(n-1)
= (1/2)(-1/2)^n-1)
= (-1)^(n-1)*(1/2)^n