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    Finding the General Term of a Sequence

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    Please provide descriptive answers to help an 8th grader learn how to approach sequencing problems. Questions and answers are provided.

    3. The 30th term of a sequence is 42. If each term in the sequence is four greater than the previous number, what is the first term?
    4. The microscopic length of a crystalline structure grows so that each day it is 1.005 times as long as the previous day. If on the third day the structure was 12.5 nm long, write a sequence for how long it is on the first several days. (nm stands for nanometer, or 1!10"9 meters.)
    5. Davis loves to ride the mini-cars at the amusement park but riders must be no more than 125 cm tall. If on his fourth birthday he is 94 cm tall and grows approximately 5.5 cm per year, at what age will he no longer be able to go on the mini-car ride?

    3. 42 ! 29(4) = !74
    4. ≈12.28, ≈12.44, 12.5, ≈12.56, ≈12.63, ...
    5. t(n) = 5.5n + 94 , so solve 5.5n + 94 !125 . n ! 5.64 . At ! 4 + 5.64 = 9.64 he will be too
    tall. Davis can continue to go on the ride until he is about 9 1
    2 years old

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    Solution Preview

    To guess what the general term of a sequence is we write a few terms and try to recognize the pattern. For this it is important that we don´t simplify the expressions we are getting but we try to write them in such a way that the pattern is clearly seen, for instance, if we are told that a certain population quadruplicates from one day to the next, then, what is the population on day n, if we start with 10 individuals?

    2nd day= 10*4,
    3rd day= (2nd day)*4=(10*4)*4=10*(4^2) and we stop there, we don´t go ahead and write it as 10*16 nor as 160, because we will not see the pattern. We need to see how the numbers that appear are related to the sequence, in this case the number 2 in the exponent of 4 equals 3-1,
    4th day=3rd day*4=10*(4^3), the ...

    Solution Summary

    Given the description of a sequence and one explicit term we find the general form of the sequence using "educated guessing" after we have written a few terms. We use that information to answer more questions about the sequences.