A mysterious sequence unfolds before us, emanating a sense of order and patterns waiting to be deciphered. As we delve into the realm of geometric sequences, we stumble upon a captivating challenge: the search for the enigmatic common ratio. With each term in this sequence revealing its own secret, we are tantalized by the possibility of unraveling the hidden code that connects them all together. Join us on this quest as we navigate the labyrinthine world of geometric sequences and embark on an intriguing journey to unlock the secrets of their common ratio.

The common ratio of a geometric sequence is the ratio between any two consecutive terms in the sequence. To find the common ratio, we can divide any term in the sequence by its preceding term. For example, let's consider the sequence:

2, 4, 8, 16, 32, ...

To determine the common ratio, we divide each term by its preceding term:

4/2 = 2, 8/4 = 2, 16/8 = 2, 32/16 = 2, ...

Therefore, the common ratio of the geometric sequence above is 2. Each term in the sequence is obtained by multiplying the preceding term by 2.

What does a geometric sequence refer to?

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

How can the common ratio of a geometric sequence be determined?

To find the common ratio of a geometric sequence, one can divide any term in the sequence by its preceding term. By doing so, we compare the ratios between consecutive terms, and if they are equal, we have found the common ratio.

Can the common ratio of a geometric sequence be negative?

Yes, the common ratio of a geometric sequence can be negative. The sign of the common ratio determines if the sequence is increasing or decreasing. A positive common ratio leads to an increasing sequence, while a negative common ratio results in a decreasing sequence.

What happens when the common ratio in a geometric sequence is less than 1?

If the common ratio in a geometric sequence is less than 1, the sequence will gradually approach zero as the terms progress. This is because each term is multiplied by a number smaller than 1, causing the values to decrease exponentially.