Common Ratio r
In mathematics, the common ratio (often denoted as “r”) is a term used in geometric sequences
In mathematics, the common ratio (often denoted as “r”) is a term used in geometric sequences.
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.
The formula to find the nth term (Tn) of a geometric sequence is:
Tn = a * r^(n-1)
Where:
– Tn is the nth term of the sequence
– a is the first term of the sequence
– r is the common ratio
– n is the position of the term in the sequence
For example, let’s consider a geometric sequence with a first term of 2 and a common ratio of 3. The first few terms of this sequence would be: 2, 6, 18, 54, 162, …
Using the formula, we can find the nth term:
Tn = 2 * 3^(n-1)
If we wanted to find the 5th term (n = 5) in this sequence, we would substitute n = 5 into the formula:
T5 = 2 * 3^(5-1)
T5 = 2 * 3^4
T5 = 2 * 81
T5 = 162
So, the 5th term of this sequence is 162.
The common ratio is an important concept in understanding geometric sequences, as it determines the growth or decay pattern of the sequence. If the common ratio is greater than 1, the terms of the sequence will increase exponentially. If the common ratio is between 0 and 1 (not inclusive), the terms of the sequence will decrease exponentially. If the common ratio is negative, the terms will alternate between positive and negative values.
Understanding the common ratio is crucial in solving problems related to geometric sequences, such as finding specific terms, finding the sum of a sequence, or identifying patterns in real-world scenarios.
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