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Question Video: Finding the General Term of a Sequence Mathematics

The general term of the sequence 3, −6, 9, −12, 15, ⋅⋅⋅ is 𝑎_𝑛 = _.

03:24

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.

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