It's just a cosmetic change, they're exactly the same sums but

they are presented slightly different.

All I've done is because I've got a n- 1 In the other series, and

I've got the range from one to to 64, if instead of n minus one I have n,

then I'm starting at zero and ending at 63.

So the idea here is that when we add up sequences incrementally,

we are creating a new sequence, the sequence of partial sums.

But because this sequence is specifically generated by adding

a consecutive number of terms of the sequence, we call it a series.

So that's what a series is.

It's the sum of the consecutive terms of a sequence.

And in some cases, we have shortcuts to find this sum.

At the start of this topic, we talked about a job offer with two payment plans.

We wrote the sequences for those and we did the sums in our spreadsheet.

Option one was we had 100 pounds in the first day and

then 50 pounds extra than the day before any new day.

An option two was a pound to start with, and then the pay was doubled every day.

So we saw option one is an arithmetic progression of common difference 50,

and first time a 100, and option two is a geometric

progression with first term one and common ratio two.

Now in terms of the arithmetic progression,

the sum of payments in terms of the arithmetic progression,

the general term is AN is 50 times n plus 1,

you can work it out from the formula we worked.

And the sum up today n, is the sum from k equals 1 to n,

of 50 times and plus 1, which is the expanded form 100

plus 150 plus 200 plus all the way to 50 times n plus 1.

For the geometric progression on payment plan two,

the general term is bn equals 2 to the power of n minus 1.

Or just 2 to the n divide by 2.

So the sum up today n is the sum

from k equals 1 to n of two

to the power of n minus 1.

So, when we looked at the spreadsheet,

which I actually show you, we had results for the sums.

So we can actually write that the sum on the thirteenth day,

which I calculated and highlighted last time.

The sum of k from 1 to 13 of ak from our

spreadsheet was 5,000 and 200,

and the sum for the second payment plan,

from 1 to 13 was 8,191.

So we can say the sum,