document.write( "Question 429173: Consider the sequence 8, 6, 3, –1, –6, ... \r
\n" ); document.write( "\n" ); document.write( "a. Find the next two terms of the sequence.
\n" ); document.write( "b. Write an explicit formula for the sequence.
\n" ); document.write( "c. Write a recursive formula for the sequence.\r
\n" ); document.write( "\n" ); document.write( "I got part a, but I'm not sure how to do the other two parts, considering that the sequence in neither arithmetic or geometric. \n" ); document.write( "
Algebra.Com's Answer #298220 by htmentor(1343)\"\" \"About 
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a. The n-th term in the series is obtained by subtracting n from the previous term. So the 6th term is -6 - 6 = -12, and the 7th term is -12 - 7 = -19.
\n" ); document.write( "c. The recursive formula which gives the n-th term from the (n-1)-th term is:
\n" ); document.write( "a_n = a_n-1 - n
\n" ); document.write( "b. To get an explicit formula, we need to find an expression which gives the n-th term without having to compute earlier terms in the sequence. Looking at the numbers, and from the recursive formula, we see that the sequence is built by subtracting n from the previous term. This is similar to the triangular number sequence 1,3,6,10,15,... which has the explicit formula a_n = n(n+1)/2. In our case we are subtracting n from the previous term, so we multiply by -1/2 instead of 1/2. However, we also need to add a constant term to reproduce the numbers of the sequence. We can write a_1 = -1(2)/2 + c = 8. Therefore, c = 9.
\n" ); document.write( "So the explict formula is:
\n" ); document.write( "a_n = -n(n+1)/2 + 9 \n" ); document.write( "
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