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Hi again! You should recognize that formula
as the sum of the first n terms
of an arithmetic series.
Let's just check what those letters mean
a is the first term in the sequence,
d is the common difference between the terms of the sequence.
The nth term therefore
is a with n minus one d's added on to it
and Sn just represents the sum of the first n terms.
So that's the formula for the sum
of the first n terms.
But where does it come from? Well let's look at what that Sn actually is.
It's the first term plus the second plus the third ...
... etcetera etcetera ...
... plus the second to last ...
plus the last. So those are
the first three and the last two in the sequence and that's what it looks like.
Now let's go vertical with that sequence.
Let's write down the first three
and the last three in that sequence -
I'm reading downwards.
Now here's the trick! Let's do the same thing again but reverse the order of them.
There's the last, the second to last, etcetera
There's the third, there's the second and there's the first.
And now we're going to add them across, add them horizontally together
and see what we get.
Well the first
pair adds up to two a's plus n minus one d's.
So does the second if you look at it carefully, and the third ...
... etcetera etcetera ...
and the last three do the same thing.
So when you add the two sequences
in reverse order
across
you get the same answer each time.
Let's add those up.
Well there are n terms in the sequence, so there's n of the those two a plus n minus one d's
So let's just summarize that. Twice the sum of the first n terms is that,
so if we just halve both sides of that equation
we get the familiar formula, the familiar equation for the sum of the first n terms
Now that's the reason why it works, it's an informal proof of it
and that's it, that's all there is to it.