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Determine whether the sequence in each question is arithmetic, geometric, or neither. Find the common difference for the arithmetic ones and the common ratio for th...

Question

Determine whether the sequence in each question is arithmetic, geometric, or neither. Find the common difference for the arithmetic ones and the common ratio for the geometric ones. Find the common difference or ratio and the 1 Oth term for each arithmetic or geometric one as appropriate.$$a_{n}=3 n-3$$

Determine whether the sequence in each question is arithmetic, geometric, or neither. Find the common difference for the arithmetic ones and the common ratio for the geometric ones. Find the common difference or ratio and the 1 Oth term for each arithmetic or geometric one as appropriate. $$a_{n}=3 n-3$$



Answers

Determine whether the sequence in each question is arithmetic, geometric, or neither. Find the common difference for the arithmetic ones and the common ratio for the geometric ones. Find the common difference or ratio and the 1 Oth term for each arithmetic or geometric one as appropriate. $$a_{n}=3 n-3$$

The unique or two three father in by would any cartoon when you'll who respectively and observed the nature off cities. If you put an equal to one, will get Penguin A to will be see the three and three upon three vehicle treatments. Three little to actually, we can notice the two worked England. This acquittal, the three very the critical are hence it is Jiffy and government. I is a little next part. We have to find some off the Christie's. The sum of 50 DerMarr. It is a by end minus one by minus. Here, a 23 accommodation is Hill three, father and and let's be minus one by this. This is

The L its term off a sequence has given us a N is equal toe end minus tree, So a one bill people to one minus tree, which is equal to minus two. It will be equal to minus three, which is equal to minus one. It really will be will do three minus three, which is equal to zero. If four will be equal to four minus three, which is equal to one on a five people do five my industry, which is equal to two on sore. So now what we seize. Let's consider the difference off the next term and the previous tone so eight to minus Avon will be will do minus one minus minus two, which is equal to one A three minus 82 will be equal toe zero minus minus one, which is equal to one. A four minus 83 will be equal to one minus zero, which is equal to one, which means that every next time is greater than the previous TEM by constant difference of one. Which means that the given sequence in is on arithmetic sequence with the constant common difference d equal to what? Now let's take the division that is a tow divided by even will be equal to minus one, divided by minus two, minus storage is equal. Do half it re divided by a two, which is equal to zero, divided by minus fan, which is equal to zero. A four divided by a tree, which is equal to one divided by zero, which is not defined on a five divided by a four will be equal toe motivated by one which is equal to do since the cautions on division are not same for any off the cases, so therefore, the sequence in is not a geometric secrets.

We have the sequence where every term could be written as three minus 2/3. And now is this arithmetic or geo metric? Let's test for arithmetic First, if this is an arithmetic sequence that it will have a common difference. D that is one term minus the previous turn, and it's really equal. No matter which two consecutive terms we click. So with this in mind, let's get started. We're going to first calculate a couple of terms that we can do our subtraction to determines in the common difference. So, first up, the first term here now that is going to be three minus 2/3 times one and so that is just equal to 7/3 as three is 3/3. Sorry, three is 9/3 and nine minutes to a sudden. Our second term is going to be three minus 2/3 times two, which is equal to three minus 4/3 9 3rd minus 4/3 5/3. And our third term is three, minus 2/3 times three 9/3 minus 6/3 for 3/3 which is one so common difference we have want for one common difference that would be one minus 5/3. And if this is an arithmetic Siri's, that should be equal to 5/3 minus 7/3. So one minus 5/3 gives us negative 2/3 and 5/3 minus 7/3 gives us negative 2/3. Those are equal. So this is in fact, an arithmetic Siri's. Now that we know it's an arithmetic serious, let's determine its common difference. Well, we've actually already done that. That's just this negative 2/3 we came up with earlier, because it is one term minus the previous, and it doesn't matter which two terms be picked. All right, now that we've done that, let's determine the some of the 1st 50 terms of this Siri's. So we have a formula once again out of the book, stating that the some of the first in terms of an arithmetic sequence is an over to times a one. The first term plus a n playing. What we have that is the sum of the 1st 50 terms is equal to 50 over to is a one well, they won. The first term, we already said, was 7/3 plus a n the term. So in this case, the 50th term which would be three minus 2/3 times 50. All right, now let's solve us 50 over to that is 25. So we have 25 times, 7/3 plus. All right. What is this equal to? Well, it's much quicker Sold in a calculator over. You could do it by hand as well. This will be 9/3 minus 50 times to our 103rd which gives us negative 91 3rd Okay, Now that we have this, we can solve this and find that is equal to 25 times seven plus 91 3rd Sorry, seven plus negative. 91 3rd which will give us just negative. 28. This does, in fact, work out to a whole number because seven plus negative 91 gives us negative 84. So now that we have this, we have 25 times negative 28 which can be quite easily solved using a calculator. This would give us negative 700. So by using a some formula from the book were able to determine that some of the 1st 50 terms in this sequence is negative. 700

We have a sequence where every term can be written as three to the end over to power. Now is this sequence geo metric or arithmetic? Let's test for geometric first. If it is geometric, then it will have what we call the common ratio that is one term divided by the previous will. Give us some constant number, that is, there will be no variables in this common ratio. Let's test this out for some generic terms, so we have generic term three to the end over to divided by the turn before this three to the and minus one. Over to now, we can use a certain export rule, which allows us to combine division into subtraction of exponents to give us three to the end over to minus and minus one. Over to then, we can combine these fractions to give us three to the n minus and plus one, that is, the negatives will cancel on this one divided by two, and the ends now cancel giving us three to the 1/2 or the square root of three. There are no ends in this term from you that it's constant, so we have in fact, found a constant common ratio, meaning this is going to be geometric and not arithmetic. So we know that our common ratio is the squared of three. That's what we just determined now, using this information, let's determine the some of the 1st 50 terms of this. So we know from the textbook that the some of the first end terms for any geometric sequence is a one. The first term. Times one minus are to the end, divided by one minus are plugging what we have. That's a one which is three to the 1/2, or the square to three times one minus the squared of three. That is our to the 50th power, divided by one minus the square root of three. Okay, so let's solve this. Well, there's not all that much that we can do by hand. The squared of three is irrational, and raising it to the 50th power is very difficult. I can't, however, we have calculators to help us out here, so I'm plugged into a calculator. This will give us the extraordinarily large number 2.0 times, 10 to the 12th power. That's a very large number. Even though We're only increasing by this word of three at a time. So that is the sum of the 1st 50 terms of this geometric secrets.


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