Pre-Calculus Sequences and Series

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 common ratio (r) can be found by dividing any term in the sequence by the previous term, i.e., r = a_n / a_(n-1).

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

The common difference (d) can be found by subtracting any term from the subsequent term, i.e., d = a_n - a_(n-1).

The sum S_n of the first n terms of a geometric series can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term and r is the common ratio.

The sum S_n of the first n terms of an arithmetic series can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where a_1 is the first term and a_n is the nth term.

The first term (a_1) is crucial as it serves as the starting point for the series and is multiplied by the common ratio to generate subsequent terms.

If the common ratio is greater than 1, the sum of the series diverges to infinity as more terms are added.

If the common ratio is between 0 and 1, the sum converges to a finite value as more terms are added.

A series converges if the sum approaches a finite limit as the number of terms increases; it diverges if the sum increases without bound.