Video Transcript
Which of the following integer
sequences does not produce any primes? Option (A) two 𝑛 plus one, option
(B) two 𝑛 plus three, option (C) two 𝑛 plus four, or option (D) two 𝑛 plus
five.
We can begin by recalling that a
prime number is a number which is divisible only by one and itself. For example, the numbers two,
three, and seven are primes. The number nine, however, would not
be prime because in addition to one and nine being factors, we know that the number
three is also a factor of nine. So let’s now combine this idea of
prime numbers with the sequences that are generated in these four given options. A sequence can be generated via an
index 𝑛. Usually this sequence starts with
integer values of 𝑛, which are one or greater. So the first term has an index of
one, the second term has an index of two, the third term has an index of three, and
so on. To find the terms of the sequence
then, we take the given 𝑛th term, or general term, and substitute in the values of
𝑛.
So let’s take this general term
that we are given in option (A) and generate the terms of the sequence. We will then be able to see if this
sequence will not produce any primes. Beginning with an index value of
one, we can find the first term in the sequence then as two times one plus one,
which gives us a value of three. The first term in this sequence is
three. Now, we could continue and
substitute in the values of two and three and so on for 𝑛. However, let’s consider the number
three. Three has only got two factors, one
and three. And that means that three is a
prime number. And therefore, if we consider “does
this sequence not produce any primes?” then the answer is no because clearly the
first term in the sequence is in fact a prime number.
Let’s repeat this process for the
𝑛th term which is given in option (B), that is, the general term two 𝑛 plus
three. The first term of this sequence
would have an index value 𝑛 equal to one. This time, we’ll have two times one
plus three, and that will give us a value of five. Let’s consider if five is a prime
number, does it have any other factors other than one and five? As it does not, then we know that
five is a prime number, and that means that we can eliminate answer option (B). We know that there will be at least
one prime number which is generated by this general term.
Let’s repeat this then for option
(C). The first term in this sequence
will be found by two times one plus four. And that gives us a value of
six. But since we know that six has more
than just two factors, we know that this is not a prime number. So let’s consider some other terms
in this sequence. The second term in this sequence
has an index 𝑛 of two. So we will be working out two times
two plus four. And that gives us an answer of
eight. And we know that eight is also not
a prime number. We can continue with this sequence,
and the third term in this sequence would be a value of 10. We might start to actually see the
pattern in this sequence which has terms of six, eight, 10, and so on. And we might then wonder what we
can say about this sequence.
We might notice that all of the
values in this sequence will be even numbers. And of course, what can we say
about prime numbers and even numbers? Well, in fact, there is only one
prime number which is also an even number, and that is the value two. We know that every other even
number except for two will have more than two factors. And because this sequence starts
with the value of six and increases, it won’t include the number two. Therefore, this sequence of two 𝑛
plus four will not produce any primes. And so we have the answer to the
question. However, let’s also consider the
final answer option.
The first term which would be
generated by the sequence with 𝑛th term two 𝑛 plus five will be two times one plus
five, and that will be equal to seven. However, we know that seven is a
prime number. And so we can’t say that this
sequence does not produce any primes. We can therefore conclude that out
of these four different options, there is only one sequence which does not produce
any primes, which is the sequence two 𝑛 plus four.
