Partial Sums

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

Example:

This is the Sequence of even numbers: {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.

Sequence

A Partial Sum is the sum of part of the sequence

 

Language Note: Partial Sums are sometimes called "Finite Series"
(a "Series" is the sum of an infinite sequence).

(Note: The sum of infinite terms is an Infinite Series.)

Sigma

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

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

 

So Sigma means to sum things up ...

Sum What?

Sum whatever appears after the Sigma:

  Sigma

so we sum n

But What is the Value of n ?

The values are shown below
and above the Sigma:

  Sigma

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:

  Sigma

Here it is in one diagram:

Sigma Notation

More Powerful

But Σ can do more powerful things than that!

We could square n each time and sum the result:

Sigma

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

Sigma

And we don't have to use n. Here we use i and sum up i × (i+1), going from 1 to 3:

Sigma

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

Sigma

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:

Sigma

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

Example:

Sigma

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

 

Adding or Subtracting Property

Here is another useful fact:

Sigma

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:

Sigma

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

Note this also works for subtraction:

Sigma

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.

Sigma   Summing 1 equals n
Sigma   Summing the constant c equals c times n
Sigma   A shortcut when summing k
Sigma   A shortcut when summing k2
Sigma   A shortcut when summing k3

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?

Sigma

Each layer is a square, so the calculation is:

12 + 22 + 32 + ... + 142

But this can be written much more easily as:

Sigma

We can use the formula for k2 from above:

Sigma

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?

Sigma

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:

Sigma

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

 

Using the "Addition Property" from above:

Sigma

Using the "Multiply by Constant Property" from above:

Sigma

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:

Sigma

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

 

And now we can use the shortcuts:

Sigma

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!