Sequences and Series A Sequence is a list of numbers connected by a rule. A Series is the sum of a sequence There are two main types of series: Arithmetic Progressions and Geometric Progressions. Arithmetic Progressions Arithmetic Progressions have a common difference between each term. nth term = a + (n - 1)d, where a = first term, and d = common difference. The sum of the first n terms is: , pairs each totalling (a + l) - (the sum of the first and last terms.) This can be rewritten using the formula for the nth term to get: Geometric Progressions Geometric Progressions have a common ratio (multiple) between each term. nth term = arn-1 , where a = first term, and r = common ratio The sum of the first n terms is: or This is the sum to infinity and this sum only converges when: The Arithmetic mean of two numbers (m and n) = The Geometric mean of two numbers =