Geometric Sequences

Front

What is a geometric sequence?

Back

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.

Front

What is the common ratio in a geometric sequence?

Back

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

Front

If the first term of a geometric sequence is 5 and the common ratio is 2, what is the second term?

Back

10

Front

How do you find the nth term of a geometric sequence?

Back

The nth term (a_n) can be found using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

Front

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

Back

The sum (S_n) of the first n terms can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

Front

Given a geometric sequence with a common ratio of 3, what is the 4th term if the first term is 2?

Back

54

Front

What happens to the terms of a geometric sequence if the common ratio is less than 1?

Back

If the common ratio is less than 1, the terms of the sequence decrease and approach zero.

Front

If a geometric sequence has a common ratio of -1, what can be said about its terms?

Back

The terms will alternate between positive and negative values.

Front

In a geometric sequence, if the first term is 10 and the common ratio is 0.5, what is the 5th term?

Back

0.3125

Front

What is the relationship between the common ratio and the growth of a geometric sequence?

Back

A common ratio greater than 1 indicates exponential growth, while a common ratio between 0 and 1 indicates decay.