Arithmetic and Geometric Sequences

An arithmetic sequence has a constant difference d between terms: [formula] Example: 3, 7, 11, 15, ... has a_1 = 3, d = 4. a_{10} = 3 + 9(4) = 39. Sum of first n terms: S_n = frac{n}{2}(a_1 + a_n) or S_n = frac{n}{2}(2a_1 + (n-1)d)

Example: Find the 20th term of 5, 8, 11, 14, ...

A geometric sequence has a constant ratio r between terms: [formula] Example: 2, 6, 18, 54, ... has a_1 = 2, r = 3. a_5 = 2 cdot 3^4 = 162. Sum of first n terms: S_n = a_1 cdot frac{1 - r^n}{1 - r} (for r neq 1)

Example: Find the 6th term of 4, 12, 36, ...