Algebra · Quick Reference

Series Cheat Sheet

Algebra II · Lesson 14/17

In one line: a series is what you get when you add up the terms of a sequence — turning a list of numbers into a running total, sometimes even an infinite one.

Key Ideas

1Series vs. Sequence. A sequence is a list of terms; a series is the SUM of those terms.

2Arithmetic Series Sum Formula. S(n) = n/2 * (a(1) + a(n)), the number of terms times the average of the first and last term.

3Geometric Series Sum Formula (finite). S(n) = a(1) * (1 - r^n) / (1 - r), for r not equal to 1.

4Infinite Geometric Series. If |r| < 1, an infinite geometric series converges to a finite sum: S = a(1) / (1 - r). If |r| >= 1, it does not converge.

5Sigma Notation. A compact way to write a sum, such as 'the sum from i=1 to n of a(i)', representing a(1)+a(2)+...+a(n).

Worked Examples

Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...

S(10) = 210

Find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, ...

S(5) = 242

Find the sum of the infinite geometric series 8 + 4 + 2 + 1 + ...

S = 16

Formulas

Arithmetic series: S(n) = n/2 * (a(1) + a(n))

Geometric series (finite): S(n) = a(1)*(1-r^n)/(1-r)

Geometric series (infinite, |r|<1): S = a(1)/(1-r)

Practice Yourself

Find the sum of the first 6 terms of 2, 4, 6, 8, ... (arithmetic).

42 (S=6/2*(2+12))

Find the sum of the first 4 terms of 1, 2, 4, 8, ... (geometric).

Find the sum of the infinite series 4 + 2 + 1 + 0.5 + ...