Geometric Series Practice

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

The sum of a finite geometric series is given by: $$S_n = \frac{a_1(1 - r^n)}{1 - r}$$ , where $$S_n$$ is the sum of the first $$n$$ terms, $$a_1$$ is the first term, $$r$$ is the common ratio, and $$n$$ is the number of terms.

The common ratio $$r$$ can be found by dividing any term by the previous term in the series: $$r = \frac{a_n}{a_{n-1}}$$ .

An infinite geometric series converges if the absolute value of the common ratio is less than 1: $$|r| < 1$$ .

The sum of an infinite geometric series is given by: $$S = \frac{a_1}{1 - r}$$ , where $$a_1$$ is the first term and $$r$$ is the common ratio.

The common ratio is 3. The sum is $$S_4 = \frac{2(1 - 3^4)}{1 - 3} = 80$$ .

If the common ratio is greater than 1, the series diverges, meaning the sum approaches infinity.

The first term of a geometric series is denoted as $$a_1$$ .

The second term is $$a_2 = a_1 \times r = 5 \times 2 = 10$$ .

A geometric series diverges if the common ratio $$r$$ is greater than or equal to 1 or less than or equal to -1.