# Partial Sums

A Partial Sum is a Sum of Part of a Sequence.

### Example:

This is the Sequence of even numbers from 2 onwards: {2, 4, 6, 8, 10, 12, ...}

This is the Partial Sum of the first 4 terms of that sequence: 2+4+6+8 = 20

Let us define things a little better now:

A Sequence is a set of things (usually numbers) that are in order.

A Partial Sum is the sum of part of the sequence.

The sum of infinite terms is an Infinite Series.

And Partial Sums are sometimes called "Finite Series".

## Sigma

Partial Sums are often written using Σ to mean "add them all up":

Σ   This symbol (called Sigma) means "sum up"

So Σ means to sum things up ...

### Sum What?

Sum whatever is after the Sigma:

Σ

n
so we sum n

### But What Values of n ?

The values are shown below
and above the Sigma:

4
Σ
n=1
n
it says n goes from 1 to 4,
which is 1, 2, 3 and 4

### OK, Let's Go ...

So now we add up 1,2,3 and 4:

4
Σ
n=1
n = 1 + 2 + 3 + 4 = 10

Here it is in one diagram:

## More Powerful

But Σ can do more powerful things than that!

We can square n each time and sum the result:
4
Σ
n=1
n2 = 12 + 22 + 32 + 42 = 30

We can add up the first four terms in the sequence 2n+1:

4
Σ
n=1
(2n+1) = 3 + 5 + 7 + 9 = 24

And we can use other letters, here we use i and sum up i × (i+1), going from 1 to 3:

3
Σ
i=1
i(i+1) = 1×2 + 2×3 + 3×4 = 20

And we can start and end with any number. Here we go from 3 to 5:

5
Σ
i=3
ii + 1 = 34 + 45 + 56

## Properties

Partial Sums have some useful properties that can help us do the calculations.

### Multiplying by a Constant Property

Say we have something we want to sum up, let's call it ak

ak could be k2, or k(k-7)+2, or ... anything really

And c is some constant value (like 2, or -9.1, etc), then:

In other words: if every term we are summing is multiplied by a constant, we can "pull" the constant outside the sigma.

### Example:

So instead of summing 6k2 we can sum k2 and then multiply the whole result by 6

Here is another useful fact:

Which means that when two terms are added together, and we want to sum them up, we can actually sum them separately and then add the results.

### Example:

It is going to be easier to do the two sums and then add them at the end.

Note this also works for subtraction:

## Useful Shortcuts

And here are some useful shortcuts that make the sums a lot easier.

In each case we are trying to sum from 1 to some value n.

Summing 1 equals n
Summing the constant c equals n times c
A shortcut when summing k
A shortcut when summing k2
A shortcut when summing k3
Also true when summing k3
Summing odd numbers

Let's use some of those:

### Example 1: You sell concrete blocks for landscaping.

A customer says they will buy the entire "pyramid" of blocks you keep out front. The stack is 14 blocks high.

How many blocks are in there?

Each layer is a square, so the calculation is:

12 + 22 + 32 + ... + 142

But this can be written much more easily as:

We can use the formula for k2 from above:

That was a lot easier than adding up 12 + 22 + 32 + ... + 142.

And here is a more complicated example:

### Example 2: The customer wants a better price.

The customer says the blocks on the outside of the pyramid should be cheaper, as they need cleaning.

You agree to:

• \$7 for outer blocks
• and \$11 for inner blocks.

What is the total cost?

You can calculate how many "inner" and "outer" blocks in any layer (except the first) using

• outer blocks = 4×(size-1)
• inner blocks = (size-2)2

And so the cost per layer is:

• cost (outer blocks) = \$7 × 4(size-1)
• cost (inner blocks) = \$11 × (size-2)2

So all layers together (except first) will cost:

Now we have the sum, let us try to make the calculations easier!

Using the "Addition Property" from above:

Using the "Multiply by Constant Property" from above:

That is good ... but we can't use any shortcuts as it is, as we are going from i=2 instead of i=1

HOWEVER, if we invent two new variables:

• j = i-1
• k = i-2

We have:

(I dropped the k=0 case, because I know that 02=0)

And now we can use the shortcuts:

After a little calculation:

\$7 × 364 + \$11 × 650 = \$9,698.00

Oh! And don't forget the top layer (size=1) which is just one block. Maybe you can give them that one for free, you are so generous!

Note: as a check, when we add the "outer" and "inner" blocks, plus the one on top, we get

364 + 650 + 1 = 1015

Which is the same number we got for the "total blocks" before ... yay!

602, 1249, 3019, 3020, 3021, 603, 1250, 8351, 8352, 8353