BitWithBite
Algebra · Quick Reference
Sequences Cheat Sheet
In one line: a sequence is an ordered list of numbers following a pattern — the two most common types, arithmetic and geometric, show up constantly throughout mathematics.
Key Ideas
1Arithmetic Sequence. Each term is found by adding a constant (the common difference, d) to the previous term.
2Arithmetic Sequence Formula. The nth term is a(n) = a(1) + (n-1)*d, where a(1) is the first term.
3Geometric Sequence. Each term is found by multiplying the previous term by a constant (the common ratio, r).
4Geometric Sequence Formula. The nth term is a(n) = a(1) * r^(n-1).
5Finding the Common Difference or Ratio. For arithmetic: subtract any term from the next one. For geometric: divide any term by the previous one.
Worked Examples
Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...
a(10) = 39
Find the 6th term of the geometric sequence 2, 6, 18, 54, ...
a(6) = 486
Find the common ratio of the sequence 100, 50, 25, 12.5, ...
r = 0.5
Formulas
Arithmetic: a(n) = a(1) + (n-1)*d
Geometric: a(n) = a(1) * r^(n-1)
Practice Yourself
Find the 5th term of 2, 5, 8, 11, ... (arithmetic).
14
Find the common difference of 10, 8, 6, 4, ...
-2
Find the 4th term of 3, 6, 12, 24, ... (geometric).
24