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What kind of sequence is the following: one, one tenth, one one hundredth, one one thousandth, and so on?
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There are two different kinds of sequences.
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It could be arithmetic or geometric.
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In an arithmetic sequence, each term is determined by adding a constant value to the previous term.
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We call that the common difference.
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So we can check if there’s a common difference.
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In a geometric sequence, each term is determined by multiplying a non-zero constant by the previous term.
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And we call that the common ratio.
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So let’s go ahead and check for a common difference and then also check for a common ratio.
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This will help determine what kind of sequence it is.
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So first, let’s look at arithmetic.
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Essentially, we’re adding something to each term to get the next one.
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So what do we keep adding by?
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What we can check by, taking a term and subtracting the previous term from it.
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So we can take one tenth minus one.
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And one is the same as 10 over 10.
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So one minus 10 on the numerator would be negative nine, and then we keep our denominator.
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So we’ve negative nine tenths.
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So now let’s check another set.
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So if we would subtract one tenth from one one hundredths, we should also get negative nine tenths, if this is arithmetic.
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One tenth is the same as ten one hundredths.
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And then we subtract the numerators, we get negative nine.
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However, the denominator is 100.
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And negative nine tenths is not the same thing as negative nine one hundredths.
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So this is not arithmetic.
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Let’s check geometric.
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So now we’re doing the exact same thing except instead of subtracting, we’re making sure that they would divide to be the same thing.
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Because if we’re multiplying each one to find what we multiplied by, you have to divide.
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One tenth divided by a one is just one tenth because dividing by one doesn’t do anything.
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Now when we take one one hundredth divided by one tenth, we actually need to multiply it by 10 over one.
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Because when you divide fractions, you multiply by the reciprocal.
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And we get ten one hundredths.
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So they’re both one tenth.
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So this sequence would be geometric only.