Question 498611: what are the next three numbers in this sequence? 0.1, -1, 10, -100
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! This is a geometric progression. Note that next term can be found by multiplying the immediately preceding term by a common ratio of -10.
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The general formula to use in finding terms of geometric progressions is:
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where:
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T is the term to be found
a is the first term in the progression
r is the common ratio or common multiplier to get from one term to the next
n is the number of the term to be found
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For this problem "a" is 0.1, r is -10, and the three values of n are 5, 6, and 7 because you are to find the 5th, 6th, and 7th terms in the series.
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That being the case, for the 5th term you can write:
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The exponent equals 4 since it is 5 - 1. Substitution results in:
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Raising -10 to the 4th power results in +10,000 and multiplying that by 0.1 gives the answer for the 5th term. The answer is:
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For the 6th term you can see that it can be found by taking the 5th term (it is 1000) and multiplying it by the common ratio (-10) to get -10,000 or you can use the equation as follows to get the same answer:
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And the 7th term can be found by multiplying the 6th term (which is -10,000) times the common ratio of -10 to get that the 7th term is +100,000. This can also be done using the equation as follows:
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Using the equation is more practical if you have to find a term for which you don't know the preceding term. What would have happened if you were told to find the 25th term of the progression instead of the next three terms.
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You could have immediately written:
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and done the calculation without having to determine all the terms from the 5th through the 24th before calculating the 25th term by multiplying the 24th term by -10.
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Hope this helps you to understand better the qualities of geometric progressions and how you can find the terms of the progression once you know the first term and the common ratio.
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