Question 1184733:
Determine the 100th term of the sequence:
1, 7, 17, 31, 49, 71,
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Previously answered; response copied below
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Technically the problem is defective; ANY subsequent numbers would form a valid sequence....
However, there is a clear pattern in the given numbers that makes it possible to find what is almost certainly the intended answer.
Look at the sequence of terms and the sequences of first and second differences:
1 7 17 31 49 71
6 10 14 18 22
4 4 4 4
The constant row of second differences means the sequence can be produced by a second degree polynomial function.
Since we need the 100th term, we need to find the quadratic function. (Alternatively, we could continue the array of numbers shown, repeating the common difference of 4 as many times as we need to reach the 100th term of the sequence. But that is not an efficient way to reach the answer!)
One way to find the quadratic function is to use the first three terms to get a system of 3 equations in 3 unknowns:
t(n)=an^2+bn+c
t(1): a+b+c=1
t(2): 4a+2b+c=7
t(3): 9a+3b+c=17
I'll let you finish the task of finding the quadratic function by that method.
NOTE: It's a good exercise in formal algebra; I strongly recommend you do it....
I will finish finding the quadratic function by a different method that you might find useful to know.
FACT: The common second difference of 4 means the quadratic function has leading coefficient 4/(2!) = 4/2 = 2. So the function is
t(n)=2n^2+bn+c
The difference between t(n) and 2n^2 will be a linear function which can easily be determined.
n t(n) 2n^2 2n^2-t(n)
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1 1 2 -1
2 7 8 -1
3 17 18 -1
The difference between t(n) and 2n^2 is the constant -1, so the quadratic function is
t(n)=2n^2-1
You can verify that by using the formula to find the given 4th, 5th, and 6th terms.
ANSWER: the 100th term of the sequence is 2(100^2)-1 = 2(10000)-1 = 19999
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In response to the student's question....
We can use a, or t, or p, or W, or whatever we want to name terms of the sequence.
I used t(n) because it represents the n-th term ("t" for term).
I specifically did not use a, because I was using a as the leading coefficient of the general quadratic function ax^2+bx+c.
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