Induction Help (1 Viewer)

Francis006

New Member
Joined
Jan 18, 2024
Messages
9
Gender
Female
HSC
2024
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

Well-Known Member
Joined
Sep 21, 2023
Messages
443
Gender
Male
HSC
2023
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.
 
Last edited:

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top