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Grade 12
Sequences & Series

Series

Understanding the sum of sequences with sigma notation - the foundation for arithmetic and geometric series.

Key Takeaways
  • A series is the sum of the terms in a sequence, denoted by $S_n$ for the sum of the first $n$ terms.
  • Sigma notation ($\Sigma$) provides a compact way to write long sums, showing the expression, index variable, and bounds.
  • Key sigma rules: you can split sums across terms, factor out constants, but bracket placement matters!
  • Understanding series notation is essential for working with arithmetic and geometric series formulas.

From Sequences to Series

You've learned how sequences are ordered lists of numbers. But what if we want to add up the terms of a sequence? That's where series come in!

Key Definition
A series is the sum of the terms in a sequence. We use the notation $S_n$ to represent the sum of the first $n$ terms.
Example:
Sequence: $1, 4, 9, 16, 25$
Series: $S_5 = 1 + 4 + 9 + 16 + 25 = 55$

Instead of writing out long sums with many plus signs, mathematicians invented a compact notation called sigma notation (using the Greek letter Σ).

Long form:
$1 + 2 + 3 + 4 + 5$
Sigma notation:
$$\sum_{i=1}^{5} i$$
Interactive Sigma Notation Explorer
Experiment with different expressions and bounds to see how sigma notation works
$$\sum_{n=1}^{5} n$$
equals: 1 + 2 + 3 + 4 + 5
Sum = 15
Term Breakdown:
n = 1
1
n = 2
2
n = 3
3
n = 4
4
n = 5
5
Finite vs Infinite Series

Series can be classified based on how many terms they contain:

Finite Series

A limited number of terms

$$\sum_{i=1}^{4} i^2$$
$= 1 + 4 + 9 + 16 = 30$
Infinite Series

Infinitely many terms

$$\sum_{i=1}^{\infty} \frac{1}{2^i}$$
$= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$
Number of Terms
For sigma notation $\sum_{i=m}^{n}$, the number of terms is $(n - m + 1)$
Example: $\sum_{i=3}^{8}$ has $8 - 3 + 1 = 6$ terms
Sigma Notation Rules
Interactive demonstrations of key sigma notation properties
Sum can be split across terms:
$$\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$$
Example:
$$\sum_{i=1}^{3} (2i + i^2)$$
$= \sum_{i=1}^{3} 2i + \sum_{i=1}^{3} i^2$
$= (2 + 4 + 6) + (1 + 4 + 9)$
$= 12 + 14 = 26$
Practice Questions
Apply your understanding of sigma notation and series
Question 1
Question 2
Question 3

Quick Self-Check Quiz

Test your understanding of series and sigma notation.

Question 1
What does $\\sum_{i=1}^{5} 2i$ equal?
Question 2
How many terms are in $\\sum_{k=4}^{10} k^2$?
Question 3
Which property allows us to write $\\sum_{i=1}^{n} (a_i + b_i) = \\sum_{i=1}^{n} a_i + \\sum_{i=1}^{n} b_i$?
Question 4
What is $\\sum_{n=0}^{3} 2^n$?
Question 5
Are $\\sum_{i=1}^{n} (3i + 1)$ and $\\sum_{i=1}^{n} 3i + 1$ the same?

What's Next?

Now that you understand series and sigma notation, you're ready to explore arithmetic and geometric series formulas.

Next: Finite Arithmetic Series →← Back to Overview