Arithmetic/Geometric Sequences & Series

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference.

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

The nth term of an arithmetic sequence can be found using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

The nth term of a geometric sequence can be found using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

The sum of the first n terms of an arithmetic series is given by: S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

The sum of the first n terms of a geometric series is given by: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.

The common difference is the constant amount that each term increases or decreases by in an arithmetic sequence.

The common ratio is the fixed number that each term is multiplied by to get the next term in a geometric sequence.

The 5th partial sum is 3110.

The first five terms are: 2, 1/2, 1/8, 1/32, 1/128.