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.