Summation Notes
is just a fancy way of
saying "1 + 2 + 5 + 10 + 17 + 26 + 37." The way to see how this works is to start by
letting I equal the first number at the bottom of the summation. In this case it is 0. Then, substitute that number into the
function part of the summation. That
gives "02 + 1" or 1.
Do the same, then, with i = 1, getting 12 + 1 or 2. Continuing this process, up to and concluding
with i = 6, gives the sum we have at the beginning.
The fact that the notation means "adding things up" is symbolized by the capital Greek letter sigma (S). The parts of the summation have meaning as shown below.
The sum at the top is not hard to evaluate. A little mental arithmetic tells you it is 98.
But suppose it had more terms. Perhaps something like 
. That won't be so
easy to do mentally. Fortunately there
are some formulas we will use to help.
You may notice that the first sum is different than the formula in the book. Since Maple starts its sums at i = 0, I am doing the same thing here. The first formula is the only one where that matters. In all the others, i = 0 gives a first term of 0 so the sum doesn't change by having i start at 0.
1) 
Notice first
that a is a constant here. So all this
becomes a + a + a + . . . + a + a. We
know that is "a times however many a's there are." We have an "a" for i = 0, one for i
= 1, and so on to i = K. that is K + 1
of them, hence the sum.
2) 
This formula
simply says if you have a constant multiplier on each term, it can be factored
out of the sum. This is the same as
saying 2x + 2y = 2 (x + y).
3) 
This formula simply
says you can rearrange the terms in an addition problem. This is the same as saying (a + b) + (c + d)
= (a + c) + (b + d).
4) 
To see why this
works, consider the case where K = 100.
In that case, we would have 1 + 2 + 3 + . . . + 98 + 99 + 100. That sum is not easy to do mentally. But let us do something with it. Write the same sum below it but in opposite
order.
1 + 2 + 3 + . . . + 98 + 99 + 100
100 + 99 + 98 + . . . + 3 + 2 + 1
Adding vertically gives 101 + 101 + 101 + . . . + 101. We would have 100 101's. That gives 100 (101). Notice that the 101 came from, among other sums, 100 + 1. But this would be twice the intended sum so we would take 100(100 + 1)/2 to get our desired answer. Now, instead of using 100, just use K and the same idea gives the indicated sum.
The remaining formulas do not lend themselves so easily to seeing why the formula works so take my word for them.
5) 
6) 
Alright, now let's go back to the one we wanted to do.
Using property 3) and 2), we have 
Now use 1) and 4) to get 