Exponential series

A geometric sequence is a type of mathematical sequence which is characterized by each term in the sequence increasing or decreasing by a certain factor. The factor is always a positive number other than 1 and is referred to as the common ratio. Geometric sequences can be used to model a variety of situations, such as population growth or the shrinking of a savings account with interest.

Given a starting term a and a common ratio r, the terms of a geometric sequence are given by the formula a ⋅ r^n. The first term in the sequence is a and the second term is a⋅r. The third term is a⋅r^2 and so on for each successive term in the sequence. Geometric sequences are finite or infinite depending on whether n is finite or infinite.

An example of a geometric sequence is the Fibonacci sequence with a = 0 and r = 1. The first few terms in this sequence are 0, 1, 1, 2, 3, 5, 8… This sequence is often used to model natural recurrence. It can be used to predict the number of petals on flowers or the number of times a pattern repeats itself in a spiral. It can also be used to model population growth if the population is not limited by resources.

Another example of a geometric sequence is a savings account with an interest rate of 5%. For a starting balance of $100, the first term is 100, the common ratio is 1.05, and the series of numbers, showing the balance of the account at the end of the year, is 100, 105, 110.25, 115.76, and so on.

In addition to being used to calculate the terms of a sequence, geometric sequences can also be used to calculate sums of an infinite sequence. If a geometric sequence with a first term a and common ratio r converges, then the sum of the sequence can be calculated using the formula a⋅(1−r^n)÷(1−r). In this formula, n is the number of terms in the sequence.

For example, to calculate the sum of the Fibonacci sequence, you would use the formula 0⋅(1−1^n)÷(1−1). This would give the result of 0, which is the sum of an infinite geometric sequence whose common ratio is 1 and whose first term is 0.

Geometric sequences are an important concept in mathematics and are used in a variety of applications, from predicting the number of petals on a flower to calculating the sum of an infinite sequence. By understanding the properties of geometric sequences, one can gain a better understanding of how to model a variety of real-world situations.

A geometric progression is a succession of numbers in which each new number is obtained by multiplying the preceding term by a common ratio. The simplest example of a geometric progression is the sequence of powers of two, where each new number is exactly twice the previous one, for example: 2, 4, 8, 16, 32, 64 and so on. There is an obvious pattern to this sequence that can be expressed mathematically as:

an = ar^n-1

where a is the first term in the sequence, and r is the constant common ratio. In the example of the powers of two, 2 is the first term, a = 2 and r = 2, so the formula to generate the sequence of powers of two becomes:

An = 2 × 2^n-1

What makes geometric progression so interesting is the fact that you can use it to calculate the terms in a sequence where the common ratio is known. Thus, by taking a few initial terms, you can work out a formula that describes the whole sequence quite easily.

The sum of the terms in a geometric sequence can also be calculated for sequences that go on forever. In such cases, the sum is known as an infinite geometric progression and can be represented by the equation:

Sn = a/(1 - r)

where a is the first term and r is the common ratio. Again, taking the powers of two example, this equation can be used to calculate the sum of the infinite sequence:

Sn = 2/(1 - 2) = 2

Finally, it is also possible to calculate the sum of a finite geometric progression. Here, the formula is given by:

Sn = a(1 - r^n)/(1 - r)

For the sequence of powers of two, a = 2, r = 2 and n = 6, so the formula can be used to calculate the sum of the first six terms in the sequence:

Sn = 2(1 - 2^6)/(1 - 2) = 63

This is the total of the six terms in the sequence: 2 + 4 + 8 + 16 + 32 + 64 = 126.

As you can see, geometric progressions are an interesting and useful mathematical concept that can be used to describe and calculate a wide range of mathematical sequences.