binäärisarjoiksi

Binäärisarjoiksi, also known as binary series, refers to a sequence of numbers where each term is obtained by doubling the previous term and then adding or subtracting a constant value. The most common form of a binary series follows the rule a_n = 2 * a_{n-1} + c, where a_n is the n-th term, a_{n-1} is the previous term, and c is a constant. Another variation can be a_n = 2 * a_{n-1} - c.

These series are distinct from geometric progressions where each term is found by multiplying the previous term by a constant ratio. In binary series, the multiplication factor is fixed at 2, but the addition or subtraction of a constant introduces a different dynamic to the sequence's growth.

For example, if the first term a_1 is 3 and the constant c is 1, the binary series would be 3, 7, 15, 31, and so on (3, 2*3+1=7, 2*7+1=15, 2*15+1=31). If the constant c were -2, starting with a_1 = 5, the series would be 5, 8, 14, 26, etc. (5, 2*5-2=8, 2*8-2=14, 2*14-2=26).

The general formula for a binary series of the form a_n = 2 * a_{n-1} + c can be expressed as a_n = c * (2^n - 1) + a_1 * 2^(n-1). This formula allows for direct calculation of any term in the series without needing to compute all preceding terms. Binäärisarjoiksi are encountered in various mathematical contexts and can be used to model certain types of growth or recursive processes.