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\n" ); document.write( "Technically the problem is defective; ANY subsequent numbers would form a valid sequence....
\n" ); document.write( "However, there is a clear pattern in the given numbers that makes it possible to find what is almost certainly the intended answer.
\n" ); document.write( "Look at the sequence of terms and the sequences of first and second differences:
\r\n" ); document.write( " 1 7 17 31 49 71\r\n" ); document.write( " 6 10 14 18 22\r\n" ); document.write( " 4 4 4 4
\n" ); document.write( "The constant row of second differences means the sequence can be produced by a second degree polynomial function.
\n" ); document.write( "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!)
\n" ); document.write( "One way to find the quadratic function is to use the first three terms to get a system of 3 equations in 3 unknowns:
\n" ); document.write( "t(n)=an^2+bn+c
\n" ); document.write( "t(1): a+b+c=1
\n" ); document.write( "t(2): 4a+2b+c=7
\n" ); document.write( "t(3): 9a+3b+c=17
\n" ); document.write( "I'll let you finish the task of finding the quadratic function by that method.
\n" ); document.write( "NOTE: It's a good exercise in formal algebra; I strongly recommend you do it....
\n" ); document.write( "I will finish finding the quadratic function by a different method that you might find useful to know.
\n" ); document.write( "FACT: The common second difference of 4 means the quadratic function has leading coefficient 4/(2!) = 4/2 = 2. So the function is
\n" ); document.write( "t(n)=2n^2+bn+c
\n" ); document.write( "The difference between t(n) and 2n^2 will be a linear function which can easily be determined.
\r\n" ); document.write( " n t(n) 2n^2 2n^2-t(n)\r\n" ); document.write( " ----------------------------\r\n" ); document.write( " 1 1 2 -1\r\n" ); document.write( " 2 7 8 -1\r\n" ); document.write( " 3 17 18 -1
\n" ); document.write( "The difference between t(n) and 2n^2 is the constant -1, so the quadratic function is
\n" ); document.write( "t(n)=2n^2-1
\n" ); document.write( "You can verify that by using the formula to find the given 4th, 5th, and 6th terms.
\n" ); document.write( "ANSWER: the 100th term of the sequence is 2(100^2)-1 = 2(10000)-1 = 19999
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\n" ); document.write( "In response to the student's question....
\n" ); document.write( "We can use a, or t, or p, or W, or whatever we want to name terms of the sequence.
\n" ); document.write( "I used t(n) because it represents the n-th term (\"t\" for term).
\n" ); document.write( "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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