Direct formula derived from recursive formula

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Homework Help Overview

The discussion revolves around deriving a direct formula from a given recursive formula T(n) = T(n-2) + n^2, with the additional condition that T(n) is constant for n < 2. Participants are exploring the implications of this recursive relationship and the nature of the differences observed in the values computed.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to compute values based on the recursive formula and notes the differences between them, suggesting a pattern. Some participants question the absence of a base case and the implications of the differences observed, while others suggest using known formulas for summation to aid in finding a direct formula.

Discussion Status

The discussion is ongoing, with participants offering hints and suggestions for approaching the problem. There is an acknowledgment of the challenges faced in deriving the direct formula, and various interpretations of the recursive relationship are being explored.

Contextual Notes

Participants note the lack of a specified base case for the recursive formula, which may influence the derivation of the direct formula. The assumption that T(n) is constant for n < 2 is also under consideration.

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Homework Statement


I have a recursive formula and I'm trying to find a direct formula.

Homework Equations


The recursive formula: T(n) = T(n-2) + n^2

No base is given. The question says "Assume that T(n) is constant for n <- 2

The Attempt at a Solution


T(1) = 0
difference: 4
T(2) = 4
difference: 6
T(3) = 10
difference: 8
T(4) = 18
difference: 10
T(5) = 28

So the difference is 2n, but then I get stuck and don't know how to go further and get a direct formula.
 
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Do you know the explicit formula for [tex]1 + 2 + \dots + n[/tex]? It will help you solve this problem, and it's something you should know anyway. (Hint: fold the sequence in the middle.)
 
The sum from k=0 to n of s+kv = (n+1) s + 1/2 n(n+1)v

But I don't know what to do because the difference doesn't start with 2
 
Imagine it did, then correct for the problem afterward!
 

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