Hi! Today we’re learning sigma notation — a fancy way of writing “add up these numbers” using the Greek letter Σ. It looks scary on first sight, but I promise it’s just a compact way of writing a long sum. Once you know how to read it, you’ll save tons of writing time, and exam questions become much easier to handle.
The Greek capital letter Σ (called “sigma”) stands for “sum”. Whenever mathematicians want to say “add up a list of things”, they use this symbol instead of writing out the whole sum.
For example, instead of writing:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
You can write the same thing as:
Both expressions mean exactly the same thing — but the sigma version is much shorter, especially when you have lots of terms. It’s just shorthand!
Why “Σ” for sum? “Σ” is the capital version of the Greek letter “sigma”, which is the equivalent of the English letter “S” — and “S” stands for Sum. Mathematicians have been using this symbol for over 250 years (since Leonhard Euler in the 1700s). Don’t let the foreign letter scare you — it’s just a fancy “S”.
Every sigma expression has 4 parts. Let me break them down using a simple example:
So this expression says: “Take (2r − 1), substitute r = 1, 2, 3, 4, 5, and add up all the answers.”
So 5Σr = 1 (2r − 1) = 25. We just turned a 5-term sum into a single number.
Memory hook: the sigma is like a recipe. The top and bottom numbers tell you how many times to repeat the recipe. The expression on the right is what to cook each time. The Σ symbol means “add it all up at the end”.
The letter inside the sum (we used r above) is called the index or counter. You can use any letter you like — most commonly r, k, or i. The letter is just a placeholder. The result is exactly the same:
4Σr = 1 r²
= 1² + 2² + 3² + 4² = 30
4Σk = 1 k²
= 1² + 2² + 3² + 4² = 30
See? Same answer. Don’t get confused if a question uses k or i instead of r — they all do the same job.
This is something students often miss. The lower limit (the bottom number) doesn’t always have to be 1. It can be 0, 2, 7 — any whole number. Always check carefully where the sum starts.
For example:
Notice that we start at k = 0 and end at k = 4 — that’s 5 terms, not 4!
Another example with a higher starting point:
This says: substitute k = 7, 8, 9, 10, 11, 12, 13, 14 (that’s 8 terms!) into (2k − 13), and add them all up.
Counting tip: the number of terms in a sigma expression is (upper limit − lower limit + 1). So 14Σk = 7 has 14 − 7 + 1 = 8 terms. Don’t forget the “+ 1” — that catches almost every student at least once!
Whenever you see a sigma expression, follow this little routine:
That’s it! With practice, you’ll be able to read sigma expressions as fast as a normal sum.
Sometimes the exam will ask you to convert a written-out sum INTO sigma notation. The trick is to find the nth term formula for the pattern, then plug it into the sigma “template”.
Let’s try one. Write the sum 2 + 4 + 6 + 8 + 10 + 12 in sigma notation.
Quick check: does 6Σk = 1 2k = 2 + 4 + 6 + 8 + 10 + 12 = 42? Yes ✓ — and 6 + 5 + 4 + 3 + 2 + 1 = 21, doubled = 42. Lovely.
Almost every modern graphing calculator (GDC) has a built-in sigma function. On a TI-84 or TI-Nspire, you can find it under the Math menu (look for “summation Σ”). On a Casio fx-CG50, it’s under the OPTN menu.
Type in the lower limit, upper limit, and expression — and your GDC will work out the sum instantly. This is the perfect way to check your answer after working it out by hand.
Evaluate: 4Σr = 1 r²
Answer:
A sequence is given by un = 2 × 3n − 1 for n ∈ ℤ⁺.
(a) Write u1 + u2 + u3 + … + u6 in sigma notation.
(b) Write u7 + u8 + u9 + … + u12 in sigma notation.
Answer:
Evaluate: 5Σk = 2 (3k + 1)
Answer:
Final word from your teacher: sigma notation is one of those things that seems intimidating but is actually just a shortcut. Once you’ve practised reading 4 or 5 expressions, you’ll start to “see through” the symbol and understand the sum behind it. In future lessons (especially arithmetic and geometric series), you’ll see sigma notation again and again — so it’s worth getting comfortable with it now. And remember: your GDC can always check your work!
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