Video Transcript
The general term of the sequence
three, negative six, nine, negative 12, 15 is 𝑎 sub 𝑛 equals what.
And we’re given four answer
options. We might notice that the terms of
this sequence alternate between positive and negative values. This type of sequence is defined as
an alternating sequence. We can take the sequence and
consider if we just had the absolute values of the sequence, then we would have the
terms three, six, nine, 12, and 15. If we took the index in this case
as 𝑛 is greater than or equal to one, then for any index 𝑛, the 𝑛th term of these
absolute values would be 𝑎 sub 𝑛 is equal to three 𝑛. But as we don’t have just three,
six, nine, 12, and so on — we have three, negative six, nine, negative 12, and so on
— then the 𝑛th term of this sequence is not three 𝑛. Furthermore, we can also say that
the 𝑛th term is not negative three 𝑛 either. In this case, the sequence would
have the values negative three, negative six, negative nine, negative 12, and so
on. However, we do have a sequence that
does very closely match three 𝑛.
So one way to find a general term
of a sequence that includes three 𝑛 but which alternates between positive and
negative is to multiply three 𝑛 by a power of negative one. We notice that options (A) and (B)
present two alternatives. Let’s have a look at the 𝑛th term
option given in (A). In order to find the first term, we
would substitute in 𝑛 is equal to one. Negative one to the power of one is
negative one, and three times one is three. Multiplying these gives us the
first term of negative three. However, if we look at the first
term in the given sequence, it’s three and not negative three. Therefore, the 𝑛th term in option
(A) is incorrect.
The 𝑛th term given in option (B)
is different because the exponent of negative one is 𝑛 plus one. When we substitute in 𝑛 is equal
to one to find the first term, we have negative one to the power of one plus one,
which is two, and negative one squared gives us one, which when multiplied by three
gives us three. This matches the given first
term. Substituting in 𝑛 is equal to two,
we get that the second term is equal to negative six. We can observe the pattern. When we have an even index, like we
did here when 𝑛 is equal to two, then the exponent of negative one will be odd. Negative one with an odd power will
give us the value of negative one. The result of this is that every
even index gives us a term value which is negative.
If we continued by substituting an
odd index of three, we would get an even value of nine. We can therefore give the answer
that it’s option (B). 𝑎 sub 𝑛 is equal to negative one
to the power of 𝑛 plus one times three 𝑛.
