The geometric explicit formula is a fundamental concept in mathematics, particularly in the fields of algebra and geometry. It is used to describe the nth term of a geometric sequence, which 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 is given by: $a_n = a_1 \cdot r^{(n-1)}$, where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.
This formula is essential in various mathematical and real-world applications, such as finance, physics, and engineering. For instance, it can be used to calculate the future value of an investment, the population growth of a city, or the distance an object travels over time. The geometric explicit formula provides a straightforward and efficient way to find any term in a geometric sequence, making it a crucial tool for problem-solving and critical thinking.
Key Points
- The geometric explicit formula is used to find the nth term of a geometric sequence.
- The formula is given by: $a_n = a_1 \cdot r^{(n-1)}$.
- The common ratio (r) is a fixed, non-zero number that is multiplied by each term to get the next term.
- The formula has various applications in mathematics, finance, physics, and engineering.
- It provides a straightforward and efficient way to find any term in a geometric sequence.
Derivation of the Geometric Explicit Formula
The derivation of the geometric explicit formula is based on the definition of a geometric sequence. Let’s consider a geometric sequence with the first term a_1 and common ratio r. The second term is obtained by multiplying the first term by r, giving a_2 = a_1 \cdot r. The third term is obtained by multiplying the second term by r, giving a_3 = a_2 \cdot r = (a_1 \cdot r) \cdot r = a_1 \cdot r^2. Continuing this pattern, we can see that the nth term is given by a_n = a_1 \cdot r^{(n-1)}.
Example Applications of the Geometric Explicit Formula
The geometric explicit formula has numerous applications in various fields. For instance, in finance, it can be used to calculate the future value of an investment. Suppose an investor deposits 1000 into a savings account with an annual interest rate of 5%. The future value of the investment after 5 years can be calculated using the geometric explicit formula: a_5 = 1000 \cdot (1 + 0.05)^4 = 1000 \cdot 1.05^4 = 1282.05. This means that the investment will be worth 1282.05 after 5 years.
| Term Number | Term Value |
|---|---|
| 1 | 1000 |
| 2 | 1050 |
| 3 | 1102.50 |
| 4 | 1157.63 |
| 5 | 1282.05 |
Geometric Sequences and Series
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. A geometric series, on the other hand, is the sum of the terms of a geometric sequence. The formula for the sum of a geometric series is given by: S_n = \frac{a_1(1 - r^n)}{1 - r}, where S_n is the sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.
Convergence of Geometric Series
A geometric series converges if the absolute value of the common ratio (|r|) is less than 1. In this case, the series converges to a finite sum, which can be calculated using the formula: S = \frac{a_1}{1 - r}. If |r| is greater than or equal to 1, the series diverges, meaning that the sum of the series approaches infinity.
What is the geometric explicit formula?
+The geometric explicit formula is given by: a_n = a_1 \cdot r^{(n-1)}, where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.
What are some applications of the geometric explicit formula?
+The geometric explicit formula has various applications in mathematics, finance, physics, and engineering, such as calculating the future value of an investment, population growth, and distance an object travels over time.
What is the difference between a geometric sequence and a geometric series?
+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. A geometric series, on the other hand, is the sum of the terms of a geometric sequence.
