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What kind of sequence is the following: 1/2, 13/6, 23/6, 11/2, 43/6 ＿?
What kind of sequence is the following: a half, 13 over six, 23 over 6, 11 over two, and 43 over six.
When a question asks what kind of sequence it is, what it actually means is: is it an arithmetic or is it a geometric sequence. So the first thing to do is actually think about well, what is an arithmetic and what is a geometric sequence. Cause if we actually work out what they are, then this is gonna help us decide what kind of sequence ours is.
So first of all, we’ll have a look at an arithmetic sequence. And the definition we’ve got for this is that it’s a sequence where the difference between two consecutive terms is constant. So e.g., it has a common difference. So now, what we’re gonna do is actually use this to work out whether our sequence is an arithmetic sequence. So what I’ve done first is actually label our terms. I’ve got 𝑛 one, 𝑛 two, 𝑛 three, et cetera. And this is just so that we can actually see which term number it is.
So now, for this to actually be an arithmetic sequence, what should actually happen is that if we actually subtract the term from the next term, then it should give us a common different throughout. So if we subtract the first term from the second term, we’re to get 13 over six minus a half which is equal to 13 over six minus three over six. Because a half is three-sixths. And we wanna have the same denominator. So this gives us a difference of 10 over six. Okay, great. So we’ve found the difference.
What we’re now gonna do is actually we’re gonna take our second term away from our third term. And when we do this, we get 23 over six minus 13 over six which again gives us 10 over six. So therefore, you can see that so far, we actually have a common difference. So, great. Well, what we’re actually gonna do is just to compare another couple of pairs, just to make sure. But it is looking like we do actually have an arithmetic sequence.
So now, if we subtract the third term from the fourth term, we would get 11 over two minus 23 over six which is gonna give us 33 over six minus 23 over six. And that’s because 11 over two is 33 over six. Because if you multiply the numerator and the denominator both by three, we’ll get 33 over six. And again, this gives us a common difference of 10 over six. So, great. So what we’re gonna do is move on to our final pair.
So finally, we’ve actually got the fifth term minus the fourth term. So we’ve got 43 over six minus 11 over two. Well, this is gonna give us 43 over six minus 33 over six. Cause as we already discussed, 11 over two is 33 over six. So again, this gives us an answer of 10 over six. So we could say yes, definitely. This sequence is an arithmetic sequence because we have a common difference of 10 over six.
So now what we’re gonna do is actually have a look at geometric sequence. Because what we’re gonna do is actually see whether it’s a geometric sequence as well. Well, a geometric sequence is a sequence where the ratio between two consecutive terms is constant. So e.g., it has a common ratio. So therefore, in order to actually work this out and see whether it is a geometric sequence, what we’re actually gonna do is actually, first of all, divide our second term by our first term.
So we’re gonna get 13 over six divided by a half. Well, to enable us to do that, what we’re gonna use is actually a rule for dividing fractions. So what we do is we actually find the reciprocal of the second fraction, so we flip it. And then, we actually multiply. So what we’ve got is 13 over six multiplied by two over one which will give us 26 over six. Because 13 multiplied by two is 26 and six multiplied by one is six. And if we actually convert this to decimal, this gives us 4.3, recurring.
So now, what we need to do is actually compare this with another pair of terms to see actually is there a common ratio. Well, if we look at the third and second terms, so we’re gonna do the third term divided by the second term, we get 23 over six divided by 13 over six. So again, using the same rule of division for fractions, we’re gonna get 23 over six multiplied by six over 13 which is gonna give us 138 over 78. Well, when we convert this to a decimal, we get 1.769, et cetera.
So therefore, we can say that actually our second term divided by our first term is not equal to our third term divided by our second term. So therefore, we do not have a common ratio. So then, if we use our definition, we could say that this cannot be a geometric sequence. So therefore, we can actually amend our answer where we said that the sequence is arithmetic, as it has a common difference of 10 over six. Because what we can say is that our sequence is only arithmetic sequence, as it has a common difference of 10 over six. But it has no common ratio.
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