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In mathematics, a common ratio is a term that appears frequently in the study of geometric sequences and series. It refers to the constant ratio between any two consecutive terms in a geometric sequence. This ratio is the same for all pairs of consecutive terms and is often denoted by the letter “r”.

To understand the concept of a common ratio, it is essential to first know what a geometric sequence is. A geometric sequence is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, consider the sequence 2, 4, 8, 16, 32, …, in which each term is obtained by multiplying the previous term by 2.

In this sequence, the first term is 2, and the common ratio is 2. To find the next term, we multiply the previous term (32) by the common ratio (2), which gives us 64. Thus, the sequence continues as 2, 4, 8, 16, 32, 64, …, where each term is obtained by multiplying the previous term by the common ratio of 2.

In general, the nth term of a geometric sequence with first term a and common ratio r can be expressed as follows:

an = ar^(n-1)

where n is the term number, and an is the nth term of the sequence.

A common ratio is a fundamental concept in the study of geometric sequences and series. One of the most important properties of a geometric sequence is that it has an infinite number of terms. This means that we can find any term of the sequence by using the formula above, no matter how large n is.

Another important property of a geometric sequence is that its terms increase or decrease exponentially. If the common ratio is greater than 1, then the terms of the sequence increase exponentially, whereas if the common ratio is between 0 and 1, then the terms of the sequence decrease exponentially. This property is due to the fact that each term is obtained by multiplying the previous term by the common ratio, which leads to an exponential growth or decay.

Geometric sequences and series are used in many areas of mathematics and science. For example, they are used in finance to model the growth of investments and the depreciation of assets. They are also used in physics to describe exponential growth and decay phenomena, such as radioactive decay and population growth.

One of the most important applications of geometric sequences and series is in the study of compound interest. Compound interest is interest that is calculated on the initial principal and also on the accumulated interest of previous periods. The formula for calculating compound interest involves the common ratio of a geometric sequence, and it is given by:

A = P(1 + r/n)^(nt)

where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

In this formula, the common ratio is (1 + r/n), which represents the factor by which the principal is multiplied in each compounding period. The formula can be used to calculate the final amount of an investment or the amount of interest earned over a given period.

In addition to compound interest, geometric sequences and series are used in many other applications. For example, they are used in computer science to analyze the performance of algorithms and data structures. They are also used in cryptography to generate random numbers and secure communication protocols.