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