Geometric Series

Front

What is a geometric series?

Back

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.

Front

What is the formula for the sum of the first n terms of a geometric series?

Back

S_n = a(1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

Front

Evaluate the geometric series: -2 - 10 - 50 - 250..., S_6

Back

-7812

Front

Evaluate the geometric series: 4 - 8 - 16 - 32..., S_7

Back

172

Front

Find the common ratio of the series: -3, -9, -27, -81,...

Back

r = 3

Front

Evaluate the geometric series: -1 - 3 - 9 - 27..., S_8

Back

-3280

Front

What is the common ratio in a geometric series?

Back

The common ratio (r) is the factor by which we multiply each term to get the next term in the series.

Front

What is the sum of the first ten terms of the sequence 4, -12, 36, -144...?

Back

-59,048

Front

If the first term of a geometric series is 5 and the common ratio is 2, what is the 4th term?

Back

The 4th term is 5 * 2^(4-1) = 5 * 8 = 40.

Front

How do you determine if a geometric series converges or diverges?

Back

A geometric series converges if the absolute value of the common ratio |r| < 1; otherwise, it diverges.