Arithmetic Series - Concept
We often use the term direct variation to describe a form of dependence of one variable on another. An equation that makes a line and crosses the origin is a form of direct variation, where the magnitude of x increases or decreases directly as y increases or decreases. Direct variation and inverse variation are used often in science when modeling activity, such as speed or velocity.
We're now going to talk about the partial sums for an arithmetic series. So basically what partial sums means is you are summing up a portion of a sequence. Okay? So remember that the sum of a sequence is called the series. So when we add up a portion of the sequence we just get what is called a partial sum, okay? So we're just adding up sum of the terms in the sequence. And we actually have a formula for the sum of an arithmetic sequence. And how it works is s of n, so this is the sum of the first n terms in this series is equal to n over 2. So that's the number of terms and then it's the s of 1 the first term plus s of n being the last term. Okay? So this is a really good formula for you to remember.
There is actually another way to write this as well, and you can either remember it which I tend not to. I have a really bad memory. Or you can derive it which is the way that I do it. Okay? So s of n is the nth term. We have an equation for that though. We have the general term which is s of n is equal to s of 1 plus n-1 times d, okay? So using substitution we could sub this right in and rewrite this equation another way.
s of n is equal to, n over 2 stays the same and what we end up with now is s of 1 plus s of 1, so we end up with 2 s of 1 plus n-1 times d. Okay? So 2 different equations to find the partial sums for an arithmetic sequence.
Kind of key things to note about which one is good for what, okay? They both need to know the number of terms you're adding up. This one is really handy when you know the first and the last term because then you can just plug it in. If you don't know the last term though and you know the difference instead this bottom one is a convenient way to use it because you don't don't have this, you can use this instead. So they're both great but they come in at different times depending on what you have available to you, okay? You could always just remember one but then you're going to have to figure things out along the way. So two different equations for summing a arithmetic series.