;;;; BEG of file "b2.soln.txt" ;;;;

Dear Numbers&Polys Enthusiast-Scholars

 Agree to use braces for grouping (not indicating sets), e.g,

	b_{n+1} means  "b sub n+1".

I'll write this solution in more detail than you needed to write on
the exam.  PLEASE GRADE my writing, and let me know (via email) of
improvements/corrections that you see.  Bear in mind that I am
limited by the format of simple email.  Best Wishes, "Prof. Jonathan"

================================================================

B2: Define a sequence b_0,b_1,b_2, ... of integers

1:	 b_0 := 0  and  b_1 := 3  and

2:	b_{n+2}  :=  7·b_{n+1} - 10·b_n
	for n=0,1, ... .

Use strong induction to prove, for all  k >= 0, that

3(k):	b_k  =  5^k  - 2^k .

 /------------------------------------------------------------\
/                                                              \
Proof:  For k=0,1,... define   R(k) := 5^k  - 2^k .

Base Cases:  Observe that

	b_0	= 0 ,		by definition,
		= 1    -  1
		= 5^0  -  2^0
		= R(0) ,	again by defn.
Similarly

	b_1	= 3 ,		by defn,
		= 5^1  -  2^1
		= R(1) ,	by defn.

================================================================
Induction case:
  Fixing a natnum n, our goal is to establish that b_{n+2} equals R(n+2)
using that (3(k)) holds for all k=0,1,2,...,n+1.    Here goes!  By defn,

	b_{n+2}	= 7·b_{n+1}		 -  10·b_n .

Courtesy equalities (3(n+1)) and (3(n)),

4:	b_{n+2}	= 7·[5^{n+1} - 2^{n+1}]	 -  10·[5^n - 2^n] .

[That was the "strong induction" step.]  Regrouping terms produces

  7·[5^{n+1} - 2^{n+1}]	=  7·5 · 5^n   -   7·2 · 2^n      and

  10·[5^n - 2^n]	=  10  · 5^n   -   10  · 2^n .

Gathering the powers of 5 in the righthand sides above thus yields

  [7·5  -  10] · 5^n	=  [35-10] · 5^n  =  25 · 5^n .

Similarly, gathering the powers of 2 above gives us

  [7·2  -  10] · 2^n	=  [14-10] · 2^n  =   4 · 2^n .

So these together with (4) say that

4':	b_{n+2}	=   5^2 · 5^n   -   2^2 · 2^n

Happily, this last equals R(n+2), which is a comfort.  QED
\                                                              /
 \____________________________________________________________/

;;;; END of file "b2.soln.txt" ;;;;