Recursive and Explicit Formulas for Sequence 1,-2,3,-4,5: Homework Help

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The sequence 1, -2, 3, -4, 5 can be expressed explicitly as a(n) = ((-1)^(n+1))n. For the recursive formula, the challenge lies in establishing a relationship between consecutive terms, indicating that a(n) depends on a(n-1) and possibly a(n-2). The discussion suggests that the recursive rule may involve a negative adjustment based on the parity of n, with the addition of 1 affecting the sign. The key is to manipulate the expression for a(n+1) to create a dependency on prior terms. Ultimately, both formulas provide a complete representation of the sequence's behavior.
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Homework Statement



I need to find the recursive and explicit formulas for the sequence 1,-2, 3, -4, 5

Homework Equations



Recursive Rule: Fn= Fn-1 + Fn-2
Explicit Rule: position of terms F1, F2, F3,...

The Attempt at a Solution



I have figured out explicit because its ((-1)^n+1)n

Recursive rule i am having trouble with because its around something like -(Fn-1 +1) or close to that
 
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Let a(n) be nth term. You should be able to say what a(1) is. Now manipulate the expression for a(n+1) until you have a right side that depends of a(n), a(n-1), etc.
 
the sign of the added one will change depdneding on whether n is odd or even
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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