Geometric Progression
Geometric progression is a mathematical concept characterized by a sequence of numbers where each term after the first is generated by multiplying the previous term by a constant known as the common ratio. This sequence can either be infinite or consist of a limited number of terms. The first term in the sequence is termed the scale factor. For example, starting with a scale factor of 3 and a common ratio of 2, the sequence progresses as 3, 6, 12, 24, and so forth. The general formula for the nth term in a geometric progression is expressed as a_n = ar^(n-1), where 'a' represents the scale factor and 'r' the common ratio.
The behavior of a geometric progression varies significantly based on the value of the common ratio. If the common ratio equals 1, the sequence remains constant. A ratio greater than 1 leads to exponential growth, while a negative ratio alternates the signs of the terms. Geometric progressions have practical applications, particularly in finance, where they can be used to calculate interest on savings accounts. For instance, a $100 deposit at a 5% interest rate produces a sequence representing the account balance over time. Overall, geometric progressions play a crucial role in various mathematical and real-world scenarios.
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Geometric Progression
The term "geometric progression" refers to a sequence of numbers that can be infinite or can have a fixed number of terms. The numbers after the first term in the sequence are determined by multiplying the previous term by a number called the common ratio. The common ratio of a geometric progression does not change throughout the entire sequence of the series. For example, an infinite geometric progression starting with the number 3 and using a common ratio of 2 would begin in this way: 3, 6, 12, 24, 48, and so on. In other words, the next term in the sequence is determined by multiplying the rightmost term by the value of the common ratio. The start value of the sequence is known as the scale factor; in the preceding example, the scale factor is 3. Thus, the general form of any geometric progression should follow the pattern of a, ar, ar2, ar3, ar4…, where a = the scale factor and r = the common ratio. This presumes that r is not equal to 0. Using these values, it is possible to calculate the nth term of a geometric progression. The formula used to determine the nth value is an = arn-1.
Overview
Geometric progressions exhibit different properties according to the value of the common ratio. If the common ratio is equal to 1, then the sequence will be constant, as in the case of the sequence 7, 7, 7, 7,…. If the common ratio is greater than 1, then the progression will exhibit exponential growth toward either positive infinity or negative infinity (the direction taken depends upon whether the scale factor had a positive sign or a negative sign). If the common ratio is equal to –1, then the progression will be a constant sequence in which terms alternate their signs from positive to negative, such as 7, –7, 7, –7,…. Similarly, if the common ratio is negative then the progression’s terms will alternate signs; if it is positive then all of the terms will also be positive.
One common use of geometric progressions that most people encounter at some point in their lives is its application for calculating interest earned on a savings account or other financial instrument. In the case of a savings account, the scale factor has a value equal to the initial deposit in the account, and the common ratio is the interest rate that money in the account earns. As with any geometric progression, the common ratio is multiplied by the rightmost term in the sequence to produce the next entry in the sequence. So, if a person deposited $100 into an account that earns 5% annually, the scale factor would be 100 and the common ratio would be 1.05. The geometric progression would be the account balance each year: 100, 105, 110.25, 115.7625, 121.550625, and so on. Geometric progression is also frequently used in making projections about the future, using the formula for calculating the nth term in the sequence.
Bibliography
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