History[ edit ] The division of history into "ages" or periods is very old, and recorded practically as early as the first development of writing. The Sumerian King List operates with dynastic regnal eras.
Site Navigation Geometric Sequences This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference.
If you need to review these topics, click here. 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. 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.
However, the recursive formula can become difficult to work with if we want to find the 50th term. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th.
This sounds like a lot of work. There must be an easier way. Rather than write a recursive formula, we can write an explicit formula. The explicit formula is also sometimes called the closed form. To write the explicit or closed form of a geometric sequence, we use anis the nth term of the sequence.
When writing the general expression for a geometric sequence, you will not actually find a value for this. 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.
You will either be given this value or be given enough information to compute it. You must substitute a value for r into the formula.
For example, when writing the general explicit formula, n is the variable and does not take on a value. But if you want to find the 12th term, then n does take on a value and it would be Your formulas should be simplified if possible, but be very careful when working with exponential expressions.
The first term in the sequence is 2 and the common ratio is 3. This is enough information to write the explicit formula.
DO NOT multiply the 2 and the 3 together. Order of operations tells us that exponents are done before multiplication. So 3 must be raised to the power as a separate operation from the multiplication. So the explicit or closed formula for the geometric sequence is.
Notice that the an and n terms did not take on numeric values. Look at the example below to see what happens. Given the sequence 2, 6, 18, 54. To find the 10th term of any sequence, we would need to have an explicit formula for the sequence.
Since we already found that in our first example, we can use it here. If we do not already have an explicit form, we must find it first before finding any term in a sequence. 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.
What happens if we know a particular term and the common ratio, but not the entire sequence? The formula says that we need to know the first term and the common ratio. We have r, but do not know a1.Every writer has to think about how to reword a sentence because revising and rephrasing is a normal part of the writing process.
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