Which Formula Can Be Used to Describe the Sequence?

There are many formulas that can be used to describe a sequence. The most common is the arithmetic sequence, which is a sequence of numbers where each successive number is obtained by adding a fixed number to the previous number. Another popular formula is the geometric sequence, which is a sequence of numbers where each successive number is obtained by multiplying the previous number by a fixed number.

A geometric sequence is a mathematical pattern in which each successive element in the sequence is multiplied by a common factor in order to obtain the next element in the sequence. Geometric sequences are characterized by a common ratio between successive terms. This common ratio may be positive, negative, or zero, and it may be a rational number or an irrational number. The study of geometric sequences has many applications in mathematical and scientific fields.

One of the simplest examples of a geometric sequence is the Fibonacci sequence, in which each successive element is the sum of the previous two elements in the sequence. The Fibonacci sequence begins with the integers 0 and 1, and the common ratio between successive terms is 1. As a result, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, and so on.

Geometric sequences arise in many different settings, both in nature and in human endeavors. For instance, the population growth of many species can be modeled by a geometric sequence. The population of a particular species of bacteria doubles every hour. If we start with 1,000 bacteria, then the population after one hour will be 2,000 bacteria, after two hours it will be 4,000 bacteria, and so on. In this case, the common ratio is 2 (that is, each successive element is twice the previous element).

Similarly, the compound interest earned on an investment can be modeled by a geometric sequence. If we start with $1,000 and earn 5% interest (compounded annually), then the value of the investment after one year will be $1,050, after two years it will be $1,102.50, and so on. In this case, the common ratio is 1.05 (that is, each successive element is 1.05 times the previous element).

Geometric sequences also arise in tessellations, which are patterns formed by repeating geometric shapes. For example, the most famous tessellation is the checkerboard pattern, which can be formed by repeating squares of equal size. Other tessellations can be formed by repeating regular polygons of different sizes, such as triangles, hexagons, or octagons. In each of these cases, the tessellation can be generated

In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with common ratio 3.

Geometric sequences are characterized by a common ratio between terms. This common ratio may be positive, negative, or zero, but it may not be a complex number. The terms in a geometric sequence increase or decrease in a predictable way, so they have specific applications in mathematical modeling.

The common ratio of a geometric sequence is the number by which each term is multiplied in order to get the next term. It is usually represented by the letter r. For example, in the sequence 2, 6, 18, 54, ..., the common ratio is 3.

Geometric sequences have a wide variety of applications in real-world situations. They can be used to model population growth, radioactive decay, and compound interest, among other things.

A important property of geometric sequences is that the terms get closer and closer to a certain number called the limit, as the sequence progresses. The limit is related to the common ratio by the following formula:

limit = a * r^n

where a is the first term in the sequence, r is the common ratio, and n is the number of terms in the sequence.

This formula allows us to find the limit of a geometric sequence even if we do not know what the common ratio is. For example, if we know that the first term in a geometric sequence is 2 and the sequence has 100 terms, we can use the formula to calculate that the limit is 2 * 3^99, which is approximately 9.97 x 10^29.

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, or common ratio. The nth term of a geometric sequence is found by multiplying the common ratio by each of the previous terms and adding them together. For example, if the common ratio is 2 and the first term is 1, then the second term would be 1*2+1=3, the third term would be 3*2+1=7, and so on. In general, the nth term of a geometric sequence can be found by the formula a_n=a_1r^{n-1}, where a_1 is the first term and r is the common ratio.

A geometric sequence is a sequence of numbers where each term after the first is multiplied by a common ratio. The common ratio can be found by dividing any two successive terms in the sequence. Finding the sum of a geometric sequence can be done using a simple formula if the common ratio is known.

The sum of a geometric sequence with first term a and common ratio r is given by:

a/(1-r)

If the common ratio is not known, it can be found by dividing any two successive terms in the sequence. Once the common ratio is known, the sum of the sequence can be found using the above formula.

For example, consider the sequence 2, 6, 18, 54. We can find the common ratio by dividing any two successive terms. For instance, 18/6=3. Therefore, the common ratio is 3. We can now plug the known values into the formula to find the sum:

2/(1-3)=2/(-2)=4

Therefore, the sum of the sequence is 4.

A geometric sequence is a sequence of numbers where each number is the product of the previous number and a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54 is a geometric sequence because each number is the previous number multiplied by 3 (the common ratio).

The common ratio of a geometric sequence can be found by dividing any two consecutive numbers in the sequence. In the example above, the common ratio is 3 because 18 divided by 6 is 3.

The starting number of a geometric sequence is called the first term. The sequence will continue indefinitely if the common ratio is between -1 and 1, exclusive. If the common ratio is greater than 1 or less than -1, the sequence will eventually reach a point where the numbers will become too large or too small and will begin to alternate between two values. This is called an oscillating sequence.

The sum of the numbers in a geometric sequence can be found using the formula:

a = a1 * (1-rn)/(1-r)

where a is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.

For example, if we wanted to find the sum of the first 10 numbers in the sequence 2, 6, 18, 54, we would use the formula above with a1=2, r=3, and n=10:

a = 2 * (1-3^10)/(1-3)

a = 2 * (1-59049)/2

a = -117147

In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54,...is a geometric sequence with common ratio 3.

The nth term of a geometric sequence is given by the formula:

a_n = a_1r^{n-1}

where a_1 is the first term in the sequence, and r is the common ratio.

For example, let's say we have the sequence 2, 6, 18, 54. We can use the formula to find the fourth term, a_4:

a_4 = a_1r^{3}

a_4 = 2(3)^{3}

a_4 = 2(27)

a_4 = 54

As we can see, the fourth term in this sequence is 54, which matches what we see in the sequence.

Using the formula, we can find the nth term of any geometric sequence. This can be useful in mathematical and scientific applications, where we might need to know the value of a term far out in a sequence.

For example, let's say we have a radioactive material with a half-life of 10 years. This means that, after 10 years, half of the material will have decayed. We can use the formula to find the amount of material remaining after 100 years:

a_100 = a_1(0.5)^{99}

a_100 = 1(0.5)^{99}

a_100 = 0.5^{99}

a_100 = 0.01

This means that, after 100 years, only 0.01 of the original material will remain. This can be useful information for scientific applications involving radioactive materials.

As we can see, the formula for the nth term of a geometric sequence is a simple one, but it can be very useful in mathematical and scientific applications.

A geometric sequence is a sequence of numbers where each term after the first is multiplied by a constant or common ratio. The sum of a geometric sequence is found by adding the terms together.

The formula for the sum of a geometric sequence is:

S = a_1(1 - r^n)/(1 - r)

where:

S is the sum of the sequence a_1 is the first term in the sequence r is the common ratio n is the number of terms in the sequence

For example, if we have a geometric sequence with a first term of 2 and a common ratio of 3, and we want to find the sum of the first 5 terms, we would use the formula like this:

S = 2(1 - 3^5)/(1 - 3)

S = 2(1 - 243)/(-2)

S = 2(-242)/(-2)

S = -482

Therefore, the sum of the first 5 terms of the geometric sequence with a first term of 2 and a common ratio of 3 is -482.

A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a constant, r. This constant is called the common ratio.

The nth term of a geometric sequence is given by:

tn = a1rn−1

where a1 is the first term in the sequence and r is the common ratio.

To determine whether a given sequence is geometric, you can use the following steps:

1) Check if there is a common ratio: To do this, simply divide any two consecutive terms in the sequence and see if you get the same number each time. If so, then there is a common ratio and the sequence is geometric.

2) Find the common ratio: Once you have confirmed that there is a common ratio, you can find it by dividing any two consecutive terms in the sequence.

3) Use the common ratio to find the nth term: Once you have the common ratio, you can use it to find the nth term in the sequence by plugging it into the formula above.

4) Check if the nth term you found matches the given term: If it does, then the sequence is geometric.

You can also use a graphing calculator or computer program to graph the sequence and see if it is a geometric sequence. If it is, then the graph will be a straight line.

A geometric sequence is a sequence of numbers where each number is the previous number multiplied by a common ratio. The common ratio is often denoted by the variable r.

The general form of a geometric sequence is:

a, ar, ar^2, ar^3, ...

where a is the first term in the sequence and r is the common ratio.

The nth term of a geometric sequence can be found using the formula:

a_n = a * r^(n-1)

The sum of the first n terms of a geometric sequence can be found using the formula:

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

There are a few properties that are common to all geometric sequences.

First, the terms in a geometric sequence are always positive if the common ratio is positive. This is because each term is simply the previous term multiplied by the common ratio. So, if the first term is positive and the common ratio is positive, then all of the terms will be positive.

Second, the terms in a geometric sequence will always be increasing if the common ratio is greater than 1. This is because each term is larger than the previous term. So, if the common ratio is greater than 1, the sequence will be increasing.

Third, the terms in a geometric sequence will always be decreasing if the common ratio is between 0 and 1. This is because each term is smaller than the previous term. So, if the common ratio is between 0 and 1, the sequence will be decreasing.

Fourth, the sum of the first n terms of a geometric sequence will always be less than the sum of the infinite series if the common ratio is less than 1. This is because the terms in the sequence get smaller and smaller as n increases. So, if the common ratio is less than 1, the sum of the first n terms will be less than the sum of the infinite series.

Finally, the sum of the first n terms of a geometric sequence will always be greater than the sum of the infinite series if the common ratio is greater than 1. This is because the terms in the sequence get larger and larger as n increases. So, if the common ratio is greater than 1, the sum of the first n terms will be greater than the sum of the infinite series.