# Find the 12th term of the geometric sequence

To create this article, 13 people, some anonymous, worked to edit and improve it over time. This article has been viewedtimes. Learn more A geometric sequence is a sequence derived by multiplying the last term by a constant.

Geometric progressions have many uses in today's society, such as calculating interest on money in a bank account. Related Articles. Author Info Last Updated: July 12, Identify the first term in the sequence, call this number a. Calculate the common ratio r of the sequence.

It can be calculated by dividing any term of the geometric sequence by the term preceding it. Identify the number of term you wish to find in the sequence. Call this number n. The n th term is given by ar n Simply insert the values of ar and n in this formula and evaluate the resulting expression to get the n th term.

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You can't do it, because that is neither a geometric nor an arithmetic sequence. Yes No.

Algebra: Evaluating Geometric Sequence

Not Helpful 24 Helpful Take this for example. Divide again by 3 and get 3. If you know the constant and one term in the geometric sequence, you can calculate any other term in the sequence. Not Helpful 18 Helpful How do I get the nth term in a geometric sequence if the first term isn't given?

If you are given any term in the sequence, treat it as if it's the first term, and proceed as usual from there. If you're not given any terms in the sequence, you cannot find the nth term. Not Helpful 13 Helpful There is no last term. Numbers are infinite, so the sequence goes on forever. Not Helpful 12 Helpful 8. You would also have to know r, the common ratio between consecutive terms.

Then use the formula for finding the sum of a geometric sequence to solve for the first term.By Yang Kuang, Elleyne Kase. If your pre-calculus teacher gives you any two nonconsecutive terms of a geometric sequence, you can find the general formula of the sequence as well as any specified term. For example, if the 5th term of a geometric sequence is 64 and the 10th term is 2, you can find the 15th term.

Just follow these steps:. You can use the geometric formula to create a system of two formulas to find r :. Plug r into one of the equations to find a 1. Now that you know a 1 and r, you can write the formula:. Thus, every year, the car is actually worth 70 percent of its value from the year before. If a 1 represents the value of a car when it was new and n represents the number of years that have passed. Notice that this sequence starts at 0, which is okay as long as the information says that it starts at 0.

Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years.

### How to Identify a Term in a Geometric Sequence When You Know Two Nonconsecutive Terms

About the Book Author Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years.Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page.

Please submit your feedback or enquiries via our Feedback page. Examples, solutions, videos, worksheets, games and activities to help Algebra II students learn about how to find the nth term of a geometric sequence. The following figure gives the formula for the nth term of a geometric sequence. Scroll down the page for examples and solutions on how to use the formula. What is the formula for a Geometric Sequence?

How to derive the formula of a geometric sequence? How to use the formula to find the nth term of geometric sequence? Geometric Sequences: A Formula for the nth Term.

This video shows how derive the formula to find the 'n-th' term of a geometric sequence by considering an example. The formula is then used to find another term of the sequence.

Show Step-by-step Solutions.He loves to write anything about education. A sequence is a function whose domain is an ordered list of numbers. These numbers are positive integers starting with 1. Sometimes, people mistakenly use the terms series and sequence.

A sequence is a set of positive integers while series is the sum of these positive integers. The denotation for the terms in a sequence is:. Finding the nth term of a sequence is easy given a general equation. But doing it the other way around is a struggle. Finding a general equation for a given sequence requires a lot of thinking and practice but, learning the specific rule guides you in discovering the general equation.

In this article, you will learn how to induce the patterns of sequences and write the general term when given the first few terms. There is a step-by-step guide for you to follow and understand the process and provide you with clear and correct computations. An arithmetic series is a series of ordered numbers with a constant difference. In an arithmetic sequence, you will observe that each pair of consecutive terms differs by the same amount. For example, here are the first five terms of the series.

Do you notice a special pattern? It is obvious that each number after the first is five more than the preceding term. Meaning, the common difference of the sequence is five.

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Usually, the formula for the nth term of an arithmetic sequence whose first term is a 1 and whose common difference is d is displayed below. Create a table with headings n and a n where n denotes the set of consecutive positive integers, and a n represents the term corresponding to the positive integers.

You may pick only the first five terms of the sequence. For example, tabulate the series 5, 10, 15, 20, 25. Solve the first common difference of a. Consider the solution as a tree diagram. There are two conditions for this step. This process applies only to sequences whose nature are either linear or quadratic. Pick two pairs of numbers from the table and form two equations. The value of n from the table corresponds to the x in the linear equation, and the value of a n corresponds to the 0 in the linear equation.

Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations. Pick three pairs of numbers from the table and form three equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.

The constant difference is 2. Since the first difference is a constant, therefore the general term of the given sequence is linear.Calculate anything and everything about a geometric progression with our geometric sequence calculator. This geometric series calculator will help you understand the geometric sequence definition so you could answer the question what is a geometric sequence? We explain the difference between both geometric sequence equations, the explicit and recursive formula for a geometric sequence, and how to use the geometric sequence formula with some interesting geometric sequence examples.

We also have built a "geometric series calculator" function that will evaluate the sum of a geometric sequence starting from the explicit formula for a geometric sequence and building, step by step, towards the geometric series formula.

The geometric sequence definition is that a collection of numbers, in which all but the first one, are obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. If you are struggling to understand what a geometric sequences is, don't fret! We will explain what this means in more simple terms later on and take a look at the recursive and explicit formula for a geometric sequence. We also include a couple of geometric sequence examples.

Before we dissect the definition properly, it's important to clarify a few things to avoid confusion. First of all, we need to understand that even though the geometric progression is made up by constantly multiplying numbers by a factor, this is not related to the factorial.

This means that the GCF is simply the smallest number in the sequence. Conversely, the LCM is just the biggest of the numbers in the sequence. Now let's see what is a geometric sequence in layperson terms.

A geometric sequence is a collection of specific numbers that are related by the common ratio we have mentioned before. The ratio is one of the defining features of a given sequence, together with the initial term of a sequence. We will see later how these two numbers are at the basis of the geometric sequence definition and depending on how they are used, one can obtain the explicit formula for a geometric sequence or the equivalent recursive formula for the geometric sequence.

Now, let's construct a simple geometric sequence using concrete values for these two defining parameters. To make things simple, we will take the initial term to be 1 and the ratio will be set to 2. The n-th term of the progression would then be. A common way to write a geometric progression is to explicitly write down the first terms.

This allows you to calculate any other number in the sequence; for our example, we would write the series as:. However, there are more mathematical ways to provide the same information. These other ways are the so-called explicit and recursive formula for geometric sequences.

Now that we understand what is a geometric sequence, we can dive deeper into this formula and explore ways of conveying the same information in fewer words and with greater precision. There exist two distinct ways in which you can mathematically represent a geometric sequence with just one formula : the explicit formula for a geometric sequence and the recursive formula for a geometric sequence.

The first of these is the one we have already seen in our geometric series example. The general formula for the n-th term is:.

## Precalculus Examples

There is another way to show the same information using another type of formula: the recursive formula for a geometric sequence. It is made of two parts that convey different information from the geometric sequence definition.

The first part explains how to get from any member of the sequence to any other member using the ratio. This meaning alone is not enough to construct a geometric sequence from scratch since we do not know the starting point. This is the second part of the formula, the initial term or any other term for that matter. Let's see how this recursive formula looks:. Where x is used to express the fact that any number will be used in its place, but also that it must be an explicit number and not a formula.

The subscript i indicates any natural number just like n but it's used instead of n to make it clear that i doesn't need to be the same number as n. Now that you know what a geometric sequence is and how to write one in both the recursive and explicit formula, it is time to apply your knowledge and calculate some stuff! With our geometric sequence calculator, you can calculate the most important values of a finite geometric sequence.

These values include the common ratio, the initial term, the last term and the number of terms. Here's a brief description of them:.In a Geometric Sequence each term is found by multiplying the previous term by a constant. Each term except the first term is found by multiplying the previous term by 2. We use "n-1" because ar 0 is for the 1st term.

Each term is ar kwhere k starts at 0 and goes up to n It is called Sigma Notation. It says "Sum up n where n goes from 1 to 4. The formula is easy to use And, yes, it is easier to just add them in this exampleas there are only 4 terms. But imagine adding 50 terms On the page Binary Digits we give an example of grains of rice on a chess board.

The question is asked:. Which was exactly the result we got on the Binary Digits page thank goodness! Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing. All the terms in the middle neatly cancel out.

Which is a neat trick. On another page we asked "Does 0. So there we have it Geometric Sequences and their sums can do all sorts of amazing and powerful things. Hide Ads About Ads. Geometric Sequences and Sums Sequence A Sequence is a set of things usually numbers that are in order. Geometric Sequences In a Geometric Sequence each term is found by multiplying the previous term by a constant. Example: 1, 2, 4, 8, 16, 32, 64, Example: 10, 30, 90,Example: 4, 2, 1, 0.

Geometric Sequences are sometimes called Geometric Progressions G. It is called Sigma Notation called Sigma means "sum up" And below and above it are shown the starting and ending values: It says "Sum up n where n goes from 1 to 4. Example: Sum the first 4 terms of 10, 30, 90,The question is asked: When we place rice on a chess board: 1 grain on the first square, 2 grains on the second square, 4 grains on the third and so on, Question: if we continue to increase nwhat happens?

Example: Calculate 0. Don't believe me?The main purpose of this calculator is to find expression for the n th term of a given sequence. Also, it can identify if the sequence is arithmetic or geometric. The calculator will generate all the work with detailed explanation.

Welcome to MathPortal. I designed this web site and wrote all the lessons, formulas and calculators. If you want to contact me, probably have some question write me using the contact form or email me on mathhelp mathportal. Math Calculators, Lessons and Formulas It is time to solve your math problem. N th term of an arithmetic or geometric sequence. Nth term of an arithmetic and geometric sequence.

Enter the first few terms of the sequence and select what to compute. You can input integers 10decimals Factoring Polynomials. Rationalize Denominator.

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