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Induction Help (1 Viewer)

Francis006

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The equation x^2+x+1=0 has roots a and b. Also, a series is defined by Tn=a^n+b^n for n=1,2,3,4...

a) Show that T1=-1 and T2= -1 (completed, but included for context)
b) Show that Tn = -Tn-1 - Tn-2 for n=3,4,5... (I have no clue from here)
c) Hence use induction to show that Tn=2cos(2nπ/3) for n=1,2,3...
d) Hence write down the value of Σn=2012 k=1 Tk
 

Luukas.2

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Part (b) is a LHs = ... and RHS = ...


Part (c) will require strong induction:
  • Prove the result for and
  • Assume it is true for and
  • Use that assumption with the definition of the recurrence, to establish the result for .​

Part (d), the sum, can likely be done with a telescoping series-type approach or by establishing the repeating pattern of values of the sequence - the latter approach will be easier in this case (in the sense of being more obvious), though the former is quicker.
 
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