# How to write an expression for the nth term of a geometric sequence

This also works for any pair of consecutive numbers. Given the sequence 2, 6, 18, 54.

## Arithmetic sequence nth term

To find the 10th term of any sequence, we would need to have an explicit formula for the sequence. Because these sequences behave according to this simple rule of multiplying a constant number to one term to get to another, they are called geometric sequences. This geometric sequence has a common ratio of 3, meaning that we multiply each term by 3 in order to get the next term in the sequence. Find the explicit formula for 0. The formula for calculating r is Site Navigation Geometric Sequences This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. If neither of those are given in the problem, you must take the given information and find them. The formula says that we need to know the first term and the common ratio. So the explicit or closed formula for the geometric sequence is. Find a6, a9, and a12 for problem 8. To find the explicit formula, you will need to be given or use computations to find out the first term and use that value in the formula. Find the recursive formula for 0. In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference. Your formulas should be simplified if possible, but be very careful when working with exponential expressions. This also works for any pair of consecutive numbers.

This will give us Notice how much easier it is to work with the explicit formula than with the recursive formula to find a particular term in a sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value.

Luckily, there is a way to arrive at the th term without the need for calculating terms 1 through If you need to review these topics, click here.

Notice that the an and n terms did not take on numeric values. But if you want to find the 12th term, then n does take on a value and it would be Find the explicit formula for a geometric sequence where and. If we match each term with it's corresponding term number, we get: n.

Since we already found that in our first example, we can use it here. This sounds like a lot of work. So the explicit or closed formula for the geometric sequence is.

## Find the 8th term of the geometric sequence

The formula says that we need to know the first term and the common ratio. This means that if we refer to the tenth term of a certain sequence, we will label it a Sequence C is a little different because it seems that we are dividing; yet to stay consistent with the theme of geometric sequences, we must think in terms of multiplication. Notice this example required making use of the general formula twice to get what we need. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th. Your formulas should be simplified if possible, but be very careful when working with exponential expressions. Site Navigation Geometric Sequences This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. And there is! The third number times 6 is the fourth number: 0. If we had to find the th term of sequence A above, we would undertake a tedious task had we decided to multiply by two each step of the way all the way to the th term. Find a6, a9, and a12 for problem 4. Rather than write a recursive formula, we can write an explicit formula.

The recursive formula for a geometric sequence is written in the form For our particular sequence, since the common ratio r is 3, we would write So once you know the common ratio in a geometric sequence you can write the recursive form for that sequence.

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