The explicit formula is also sometimes called the closed form.
For example, when writing the general explicit formula, n is the variable and does not take on a value. If we do not already have an explicit form, we must find it first before finding any term in a sequence. The formula says that we need to know the first term and the common ratio.
To find the 10th term of any sequence, we would need to have an explicit formula for the sequence. So the explicit or closed formula for the geometric sequence is. If we simplify that equation, we can find a1. In this situation, we have the first term, but do not know the common ratio.
This is enough information to write the explicit formula. 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.
However, we have enough information to find it. You will either be given this value or be given enough information to compute it. For a review of how to prove that a squence is arithmetic, read through the Supplementary Reading on that topic.
What happens if we know a particular term and the common ratio, but not the entire sequence? Your formulas should be simplified if possible, but be very careful when working with exponential expressions.
We have r, but do not know a1.
We often symbolize this constant ratio by r. This sequence is neither arithmetic or geometric but the numerator and denominator are!
Here is another example of writing a series in summation notation. As always, if you have questions about what is written here, if you find mistakes, or if you have comments, please send them to me. So 3 must be raised to the power as a separate operation from the multiplication.
Site Navigation Geometric Sequences This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence.
The first term in the sequence is 2 and the common ratio is 3. Now that we know the first term along with the r value given in the problem, we can find the explicit formula. Find the explicit formula for 0.
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.
Knowing that the sequence is arithmetic allows us to use the pattern of an arithmetic sequence in order to find the general term. Find the recursive formula for 0. Since we are told that the sequence is geometric we know that the ratio of ANY two consecutive terms is a constant, r.Watch video · So if you wanna figure out the th term of this sequence, I didn't even have to write it in this general term, you could just look at this pattern.
It's going to be, and I'll do it in pink, the th term in our sequence. we write the general (or nth) term of a geometric sequence as: a n = a 1 r n - 1 This is the formula that will be used when we find the general (or nth) term of a geometric sequence.
Oct 19, · Geometric Sequences: A Formula for the' n - th ' Term.
Given two terms find the nth term of a geometric sequence How To Write an Explicit Formula For Arithmetic & Geometric Sequences With. Geometric Sequences: A Formula for the’ n – th ‘ Term.
Topic: Calculus, Sequences and Series. Tags: formula, geometric, geometric sequence. Related Math Tutorials: Geometric Sequences: A Quick Intro; Arithmetic Sequences: Finding a General Formula Given Two Terms; Geometric Series and the Test for Divergence; Geometric Series and the Test.
Geometric Sequence Calculator Find indices, sums and common ratio of a geometric sequence step-by-step. Equations. General; Arithmetic; Geometric; sequences. geometric sequence. What I want to Find. Common Ratio Next Term N-th Term Value given Index Index given Value Sum. Please pick an option first.
What is Given. To write the explicit or closed form of a geometric sequence, we use a n is the nth term of the sequence. When writing the general expression for a geometric sequence, you will not actually find a value for this.Download