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 * (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.

Front

How do you find the common ratio in a geometric series?

Back

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

Front

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

Back

S_6 = -7812.

Front

Find S_6 for the geometric series: (-4/5) + 8 + (-80) + 800 + ....

Back

S_6 = 72,727.2.

Front

What is the total distance traveled by a bouncing ball dropped from 25 cm after five bounces if it reaches heights of 16 cm, 12.8 cm, and 10.24 cm?

Back

Total distance = 84.04 cm.

Front

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

Back

Sum = -59,048.

Front

What is the first term of a geometric series?

Back

The first term (a_1) is the initial value from which the series begins.

Front

What is the significance of the common ratio in a geometric series?

Back

The common ratio determines how quickly the terms of the series grow or shrink.

Front

How does a geometric series differ from an arithmetic series?

Back

In a geometric series, each term is multiplied by a constant (common ratio), while in an arithmetic series, each term is added to a constant (common difference).