Sums and Products

Summation notation writes repeated addition compactly.

The symbol $\sum$ means "sum".

Summation notation compresses repeated addition.

A Small Sum

This expression

\sum_{i=1}^{3} x_i

means:

add x_1, x_2, and x_3

So if

x_1 = 2,\quad x_2 = 4,\quad x_3 = 6

then:

\sum_{i=1}^{3} x_i = 2 + 4 + 6 = 12

MATH-C01-T06-001Exercise: Compute a short sum

If $x_1 = 3$ , $x_2 = 4$ , and $x_3 = 5$ , what is $\sum_{i=1}^{3} x_i$ ?

Your answer

Compute it first, then check your number.

HintExpand first

Write the sum as $x_1 + x_2 + x_3$ .

SolutionExpansion

Expand the summation:

\sum_{i=1}^{3} x_i = x_1 + x_2 + x_3 = 3 + 4 + 5 = 12

The first equality expands the notation. The second substitutes the given values. Keeping those steps separate prevents many mistakes.

Reading the Parts

\sum_{i=1}^{n} x_i

read the parts like this:

i = 1        start with i equal to 1
n            stop at n
x_i          add the i-th value of x

The index $i$ is a counter. It moves from the starting value to the ending value.

The counter is temporary. After the sum is complete, $i$ has done its job.

MATH-C01-T06-002Exercise: Count the terms

How many terms are in this sum?

\sum_{i=1}^{5} x_i

Your answer

Compute it first, then check your number.

HintList the counter values

The counter $i$ starts at 1 and stops at 5.

SolutionCounter values

The counter takes the values:

1,\ 2,\ 3,\ 4,\ 5

so there are 5 terms. The upper limit is included because the sum runs from 1 through 5.

Product Notation

The symbol $\prod$ means repeated multiplication.

For example:

\prod_{i=1}^{3} x_i = x_1 x_2 x_3

x_1 = 2,\quad x_2 = 3,\quad x_3 = 4

then:

\prod_{i=1}^{3} x_i = 2 \cdot 3 \cdot 4 = 24

Product notation appears less often at the beginning than sums, but it is useful in probability.

MATH-C01-T06-003Exercise: Compute a product

If $x_1 = 2$ , $x_2 = 3$ , and $x_3 = 4$ , what is $\prod_{i=1}^{3} x_i$ ?

Your answer

Compute it first, then check your number.

HintExpand first

Write the product as $x_1x_2x_3$ .

SolutionMultiplication

\prod_{i=1}^{3} x_i = x_1x_2x_3 = 2 \cdot 3 \cdot 4 = 24

Product notation works like summation notation, but the repeated operation is multiplication instead of addition.

Why Sums Matter in ML

Sums appear everywhere:

dot products
averages
losses over a dataset
probabilities over possible outcomes
gradients that collect many contributions

For example, an average loss often has this shape:

\frac{1}{n}\sum_{i=1}^{n} \ell_i

Read it as:

add the loss for each example
divide by the number of examples

MATH-C01-T06-004Exercise: Average loss

Three examples have losses $2$ , $5$ , and $8$ .

What is the average loss?

Your answer

Compute it first, then check your number.

HintUse the average pattern

First compute $2 + 5 + 8$ , then divide by 3.

SolutionAverage

\frac{2 + 5 + 8}{3} = \frac{15}{3} = 5

The numerator is the total loss across the three examples. Dividing by 3 turns that total into an average.

Next, we turn these habits into a practical method: reading formulas aloud.