QuestionDetermine the formula for using the sequence given as .
Studdy Solution
STEP 1
Assumptions1. The sequence is . The given values of the sequence are
STEP 2
Observe the pattern in the sequence. The sequence alternates between negative and positive numbers.
STEP 3
Also note that the absolute value of each term in the sequence increases by1 each time.
STEP 4
We can see that the sequence can be split into two sub-sequences one for even-indexed terms and one for odd-indexed terms.
STEP 5
The even-indexed terms are and their values are
STEP 6
The odd-indexed terms are and their values are
STEP 7
The even-indexed terms form an arithmetic sequence with a common difference of .
STEP 8
The odd-indexed terms form an arithmetic sequence with a common difference of .
STEP 9
We can express the nth term of an arithmetic sequence as , where is the first term and is the common difference.
STEP 10
For the even-indexed terms, the first term and the common difference . So, we can express the nth term of the even-indexed sequence as .
STEP 11
implify the expression for .
STEP 12
For the odd-indexed terms, the first term and the common difference . So, we can express the nth term of the odd-indexed sequence as .
STEP 13
implify the expression for .
STEP 14
We can combine the expressions for and into a single expression for using the piecewise function.
This is the expression for based on the values of
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