Section 9.3

Summation Notation

Sigma notation is the standard way to write series on Exam 4 — you'll see it in arithmetic and geometric sum problems, and it comes back in the Binomial Theorem. Getting comfortable reading and manipulating $\Sigma$ now means those problems won't slow you down.

⚡ Quick Summary

Summation notation is just a shorthand for "add up a bunch of things." The big Greek letter $\Sigma$ (sigma) means "sum." Instead of writing out $1 + 4 + 9 + 16 + 25$ , you can write $\sum_{i=1}^{5} i^2$ — which says "plug in $i = 1, 2, 3, 4, 5$ into $i^2$ and add up the results." It's like a compact instruction: the bottom tells you where to start, the top tells you where to stop, and the expression on the right tells you what to compute each time.

Sigma Notation

\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n

i is the index, m is the start, n is the end, aᵢ is the expression

Sum of a Constant

\sum_{i=1}^{n} c = c \cdot n

Adding the same number n times

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Sigma notation follows a few handy rules that let you break big sums into smaller, easier pieces. You can split a sum apart — $\sum(a_i + b_i)$ becomes $\sum a_i + \sum b_i$ . You can pull a constant multiplier out front — $\sum 3i$ becomes $3 \cdot \sum i$ . These rules work just like distributing in regular algebra.

The real time-saver comes from shortcut formulas. Instead of adding $1 + 2 + 3 + \cdots + 100$ one by one, you can use $\frac{n(n+1)}{2}$ and get 5050 instantly. There's a similar formula for the sum of squares. With these formulas plus the splitting rules, you can evaluate large sums without doing hundreds of additions.

One thing to watch: the shortcut formulas assume you start at $i = 1$ . If your sum starts at $i = 0$ or $i = 2$ , you need to adjust — either handle the extra/missing terms separately or shift the formula. When the upper limit is small (like 4 or 5), it's often faster to just plug in each value and add by hand rather than use the formulas.

Key Formulas

Sigma Notation

\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n

i is the index, m is the start, n is the end, aᵢ is the expression

Sum of a Constant

\sum_{i=1}^{n} c = c \cdot n

Adding the same number n times

Sum of First n Positive Integers

\sum_{i=1}^{n} i = \frac{n(n+1)}{2}

The classic Gauss formula: 1 + 2 + 3 + ⋯ + n

Sum of First n Squares

\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

Linearity of Summation

\sum_{i=1}^{n}(a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i, \qquad \sum_{i=1}^{n} c \cdot a_i = c \cdot \sum_{i=1}^{n} a_i

Split sums apart and pull constants out — just like distributing

Key Takeaways

✓Three formulas to memorize: $\sum c = cn$ , $\sum i = \frac{n(n+1)}{2}$ , $\sum i^2 = \frac{n(n+1)(2n+1)}{6}$
✓Linearity: split sums apart and pull constants out — $\sum(a_i + b_i) = \sum a_i + \sum b_i$
✓Always check the starting index — formulas assume $i = 1$ ; adjust if the sum starts at $0$ or $2$
✓For small upper limits (4 or 5), just expand and add — it's faster than using formulas

⚠️ Common Mistakes

✗Forgetting to check the starting index — most closed-form formulas assume $i = 1$ , so if your sum starts at $i = 0$ or $i = 2$ , you need to adjust or compute the extra/missing terms separately
✗Trying to pull out a variable as if it were a constant — $\sum 3i^2$ lets you pull out the $3$ , but $\sum i \cdot i^2$ does NOT let you pull out the $i$ because $i$ changes with each term
✗Mixing up the closed-form formulas — $\sum i = \frac{n(n+1)}{2}$ vs. $\sum i^2 = \frac{n(n+1)(2n+1)}{6}$ ; double-check which one you need before plugging in
✗Forgetting that $\sum_{i=1}^{n} c = cn$ , not just $c$ — a constant summed $n$ times gives $cn$

Next: Section 11.4 Builds on: 11.2 Builds on: 11.3

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11.3 Geometric Sequences

11.4 Series & Their Notations