Please answer the 4th question. I. (2 points) Let F, be the Fibonacci numbers, w

Question

Please answer the 4th question.

I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points) Use induction to prove that fr all natural numbersas divisible by x-1. Heileman, p.415) 3. (2 points) Prove by induction that l 2 + 22 + 32 + + n2 4. (2 points) Prove by induction that the sum of the first n odd positive integers is n, i.e., 1 +3+5 (2n)-r 5, (2 points) Prove that -ok2k-(n-1)2n+1 + 2. (Heileman, p. 415) 6. (2 points) Assuming a and b are arbitrary constants, and 0

Explanation / Answer

Hi,
Proof by induction as 3 steps,
1. Prove for n=1
2. Assume true for n=k
3. Then prove for n=k+2, using above assumption,
then we are done,
Now, given claim is 1+3+..+(2n-1)=n2
step1: For n=1, LHS=1, RHS=12=1, hence true,
step2: Assume true for n=k i.e 1+3+..+(2k-1)=k2
step3: For n=k+1.
LHS= 1+3+...+(2k-1)+(2(k+1)-1)
  but we know from above assumption, the sume of k terms,substituting, we get
=k2+(2(k+1)-1)
  = k2+2k+1
   = (k+1)2
therefore LHS=RHS
hence proved
Thumbs up if this was helpful, otherwise let me know in comments

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