In a geometric sequence, the ratio between consecutive terms is
The Correct Answer and Explanation is:
In a geometric sequence, the ratio between consecutive terms is called the common ratio (denoted as r). This ratio is constant, meaning that each term is found by multiplying the previous term by the same factor. To better understand this, let’s break down the structure of a geometric sequence.
A geometric sequence is expressed as:a1,a2,a3,a4,…a_1, a_2, a_3, a_4, \dotsa1,a2,a3,a4,…
Where:
- a1a_1a1 is the first term
- a2a_2a2 is the second term
- a3a_3a3 is the third term, and so on.
The relationship between any two consecutive terms can be described by the common ratio r. Specifically, the ratio between the second term and the first term is:r=a2a1r = \frac{a_2}{a_1}r=a1a2
Likewise, the ratio between the third term and the second term is:r=a3a2r = \frac{a_3}{a_2}r=a2a3
For a sequence to be geometric, this ratio must be constant throughout. This means:a2a1=a3a2=a4a3=…\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \dotsa1a2=a2a3=a3a4=…
The common ratio r can be any real number, including positive, negative, or even a fraction. For example, in the sequence 2,6,18,54,…2, 6, 18, 54, \dots2,6,18,54,…, the common ratio is:r=62=3r = \frac{6}{2} = 3r=26=3
Thus, each term is obtained by multiplying the previous term by 3.
In contrast, if the sequence involves a negative common ratio, such as 5,−10,20,−40,…5, -10, 20, -40, \dots5,−10,20,−40,…, the ratio r is -2. In this case, each term alternates in sign, but the absolute values grow by a factor of 2.
In summary, the common ratio r is the constant factor by which each term in a geometric sequence is multiplied to obtain the next term. This makes the sequence predictable and consistent.
