Question Video: Finding the General Term of a Given Sequence Mathematics

Find, in terms of 𝑛, the general term of the sequence (1, 2, 4, 8, ...) where 𝑛 β‰₯ 1.

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Video Transcript

Find in terms of 𝑛 the general term of the sequence one, two, four, eight, and so on where 𝑛 is greater than or equal to one.

In this question, we’re asked to find the general or 𝑛th term of the sequence. So let’s begin by establishing what type of sequence we have.

If there is a common difference where the same value is added or subtracted between any two terms, then we would have an arithmetic sequence. Therefore, let’s see if we can calculate the difference between the first and second terms. Well, we can get to the second term from the first term by adding one. However, to go from the second to the third term, we must add two. And from the third to the fourth term, we must have added four. Therefore, this is not an arithmetic sequence.

The next thing we might choose to do is to establish if we have a geometric sequence. In a geometric sequence, there is a fixed ratio between successive terms. Let’s calculate the ratio between the first and second terms. And we can easily see that if we multiply the first term, one, by two, we would get the second term of two. We could also get from the second term to the third term by multiplying by two. And finally, we can see that the ratio between the third term and the fourth term is also two, which means that we do indeed have a geometric sequence.

We can remember that the 𝑛th term π‘Ž sub 𝑛 of a geometric sequence is calculated as π‘Ž times π‘Ÿ to the power of 𝑛 minus one, where π‘Ž or π‘Ž sub one is the first term and π‘Ÿ is the fixed ratio between terms. In this sequence, we have calculated that the ratio between terms is two. So that means that π‘Ÿ equals two. The value of π‘Ž is simply the first term in the sequence, which is one. When we substitute these values into the 𝑛th term formula, we get π‘Ž sub 𝑛 is equal to one times two to the power of 𝑛 minus one. We can then simplify this answer to give the general term of the sequence one, two, four, eight, and so on as to two the power of 𝑛 minus one.

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