Consider the following geometric sequence:
360, 36, 3.6, 0.36, ...
What is the explicit formula for this sequence?
To determine the explicit formula for the given sequence \(360, 36, 3.6, 0.36, \dots\), we first identify the type of sequence. Each term is obtained by multiplying the previous term by a constant value. For instance, \(36 / 360 = 0.1\) and \(3.6 / 36 = 0.1\). Thus, this is a geometric sequence. The standard form for the n-th term of a geometric sequence is \(a_n = a \cdot r^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.
From the sequence provided, the first term \(a\) is \(360\). The common ratio \(r\) is found by dividing any term by its preceding term:
\[r = \frac{36}{360} = 0.1\]
We can verify this ratio with the subsequent terms:
\[36 \times 0.1 = 3.6\]
\[3.6 \times 0.1 = 0.36\]
Since the common ratio is consistent, we have \(a = 360\) and \(r = 0.1\).
Substituting the values \(a = 360\) and \(r = 0.1\) into the general formula \(a_n = a \cdot r^{n-1}\), we obtain the explicit formula for the sequence:
\[a_n = 360 \cdot (0.1)^{n-1}\]
This formula allows us to find any term in the sequence by providing the term number \(n\). The first part of the expression is 360 and the second part is 0.1.
Geometric sequences are used in various fields, such as finance for calculating compound interest or in biology for modeling population growth. In this specific sequence, it can represent a process of decay or depreciation, like the value of a piece of equipment decreasing by 90% each year, leaving only 10% of its previous value. Understanding these formulas helps in predicting future values or identifying trends in rapidly changing data.