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## Arithmetic & Geometric Sequences

Find the sum of the infinite geometric series given by:. Before we do anything, we'd better make sure our series is convergent. Otherwise, we can't find an infinite sum at all. What's our common ratio? Now we can nab our infinite geometric series formula. Therefore the sum is infinite. Hey, that means we're done! Find the sixth partial sum of the geometric series given by:. Use the formula for the partial sum of a geometric series. You've got it printed out on a little card in your wallet, right? Find the sum:. Now pop in the first term a 1 and the common ratio r. Log In. Sequences and Series. Substitute for a 1 and r in the formula and watch the magic unfold. Show Next Step. Example 2. Let's hunt down that common ratio. It's just the number in parentheses. Example 3. Substitute the values you know into the formula. Now slog through the actual math and simplify everything as much as you can. Example 4. It's our best bud. Rock those fractions. Logging out…. Logging out You've been inactive for a while, logging you out in a few seconds I'm Still Here! W hy's T his F unny?

## Geometric Sequences Intro Examples Arith. Series Geo. The two simplest sequences to work with are arithmetic and geometric sequences. An arithmetic sequence goes from one term to the next by always adding or subtracting the same value. For instance, 2, 5, 8, 11, 14, The number added or subtracted at each stage of an arithmetic sequence is called the "common difference" dbecause if you subtract that is, if you find the difference of successive terms, you'll always get this common value. A geometric sequence goes from one term to the next by always multiplying or dividing by the same value. So 1, 2, 4, 8, 16, The number multiplied or divided at each stage of a geometric sequence is called the "common ratio" rbecause if you divide that is, if you find the ratio of successive terms, you'll always get this common value. To find the common difference, I have to subtract a successive pair of terms. It doesn't matter which pair I pick, as long as they're right next to each other. To be thorough, I'll do all the subtractions:. They gave me five terms, so the sixth term of the sequence is going to be the very next term. I find the next term by adding the common difference to the fifth term:. To find the common ratio, I have to divide a successive pair of terms. To be thorough, I'll do all the divisions:. They gave me five terms, so the sixth term is the very next term; the seventh will be the term after that. To find the value of the seventh term, I'll multiply the fifth term by the common ratio twice:. Since arithmetic and geometric sequences are so nice and regular, they have formulas. For arithmetic sequences, the common difference is dand the first term a 1 is often referred to simply as " a ". Since we get the next term by adding the common difference, the value of a 2 is just:.

## Sequences and Series In this section we are going to take a brief look at three special series. Actually, special may not be the correct term. However, notice that both parts of the series term are numbers raised to a power. This means that it can be put into the form of a geometric series. We will just need to decide which form is the correct form. It will be fairly easy to get this into the correct form. This can be done using simple exponent properties. We can now do some examples. However, this does provide us with a nice example of how to use the idea of stripping out terms to our advantage. From the previous example we know the value of the new series that arises here and so the value of the series in this example is. However, we can start with the series used in the previous example and strip terms out of it to get the series in this example. We will strip out the first two terms from the series we looked at in the previous example. We can now use the value of the series from the previous example to get the value of this series. Consider the following series written in two separate ways i. This is now a finite value and so this series will also be convergent. In other words, if we have two series and they differ only by the presence, or absence, of a finite number of finite terms they will either both be convergent or they will both be divergent. The difference of a few terms one way or the other will not change the convergence of a series. In this portion we are going to look at a series that is called a telescoping series. The name in this case comes from what happens with the partial sums and is best shown in an example. By now you should be fairly adept at this since we spent a fair amount of time doing partial fractions back in the Integration Techniques chapter. If you need a refresher you should go back and review that section. So, what does this do for us? Notice that every term except the first and last term canceled out. This is the origin of the name telescoping series. This also means that we can determine the convergence of this series by taking the limit of the partial sums. In telescoping series be careful to not assume that successive terms will be the ones that cancel. Consider the following example. The partial sums are. In this case instead of successive terms canceling a term will cancel with a term that is farther down the list. The end result this time is two initial and two final terms are left. So, this series is convergent because the partial sums form a convergent sequence and its value is.

## How to Recognize a P-Series 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.

## Geometric Series

Intro Examples Arith. Series Geo. You can take the sum of a finite number of terms of a geometric sequence. Note: Your book may have a slightly different form of the partial-sum formula above. All of these forms are equivalent, and the formulation above may be derived from polynomial long division. I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of —2. Plugging into the summation formula, I get:. The notation " S 10 " means that I need to find the sum of the first ten terms. Dividing pairs of terms, I get:. Unlike the formula for the n -th partial sum of an arithmetic series, I don't need the value of the last term when finding the n -th partial sum of a geometric series. So I have everything I need to proceed. When I plug in the values of the first term and the common ratio, the summation formula gives me:. I will not "simplify" this to get the decimal form, because that would almost-certainly be counted as a "wrong" answer. Instead, my answer is:. Note: If you try to do the above computations in your calculator, it may very well return the decimal approximation of As you can see in the screen-capture above, entering the values in fractional form and using the "convert to fraction" command still results in just a decimal approximation to the answer. But really! Take the time to find the fractional form. They've given me the sum of the first four terms, S 4and the value of the common ratio r. Since there is a common ratio, I know this must be a geometric series. Plugging into the geometric-series-sum formula, I get:. Then, plugging into the formula for the n -th term of a geometric sequence, I get:.