6-7 Flashcardizz Review - Geometric 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 nth term of a geometric sequence can be found using the formula: $$a_n = a_1 \times r^{(n-1)}$$ , where $$a_1$$ is the first term, $$r$$ is the common ratio, and $$n$$ is the term number.

The common ratio is 2, as each term is multiplied by 2 to get the next term.

The sum of the first n terms of a geometric series can be calculated using the formula: $$S_n = a_1 \times \frac{1 - r^n}{1 - r}$$ , where $$a_1$$ is the first term and $$r$$ is the common ratio.

The sum is $$S_4 = 2 \times \frac{1 - 3^4}{1 - 3} = 2 \times \frac{1 - 81}{-2} = 2 \times 40 = 80$$ .

A geometric series is the sum of the terms of a geometric sequence.

Yes, the sequence is geometric with a common ratio of 3.

The 5th term is 324.

The sum is $$S_5 = 1 \times \frac{1 - 2^5}{1 - 2} = 31$$ .

The common ratio is $$r = \frac{27}{81} = \frac{1}{3}$$ .