Which sequence shows a pattern where each term is 1.5? This is a question that often puzzles students and enthusiasts alike in the field of mathematics. In this article, we will explore a specific sequence that exhibits this unique pattern, and delve into the underlying mathematical principles that govern it.
In mathematics, a sequence is an ordered list of numbers that follow a specific pattern. Each number in the sequence is called a term, and the pattern is defined by the relationship between consecutive terms. In the case of a sequence where each term is 1.5 times the previous term, we are dealing with a geometric sequence. Geometric sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including finance, physics, and computer science.
A geometric sequence is characterized by a constant ratio between consecutive terms. In this case, the ratio is 1.5, which means that each term is 1.5 times the term that precedes it. To illustrate this, let’s consider the following sequence:
1, 1.5, 2.25, 3.375, 5.0625, …
As we can observe, each term in this sequence is 1.5 times the term that came before it. For example, the second term (1.5) is 1.5 times the first term (1), the third term (2.25) is 1.5 times the second term (1.5), and so on. This pattern continues indefinitely, resulting in an infinite sequence.
The general formula for a geometric sequence is given by:
a_n = a_1 r^(n-1)
where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.
In our example, the first term (a_1) is 1, and the common ratio (r) is 1.5. Therefore, the formula for our sequence becomes:
a_n = 1 1.5^(n-1)
This formula allows us to calculate any term in the sequence by plugging in the desired term number (n). For instance, to find the 10th term, we would use the following calculation:
a_10 = 1 1.5^(10-1) = 1 1.5^9 = 1 14,348.3843 ≈ 14,348.3843
In conclusion, the sequence that shows a pattern where each term is 1.5 is a geometric sequence with a common ratio of 1.5. This type of sequence is widely used in various mathematical applications and is a fundamental concept in the field of mathematics. By understanding the properties and formula of geometric sequences, we can better appreciate the beauty and complexity of this fascinating mathematical concept.
