Read..
http://en.wikipedia.org/wiki/Mathematical_induction
Variants
Induction basis other than 0 or 1
1. Showing that the statement holds when *n* = *b*.
2. Showing that if the statement holds for *n* = *m* ≥ *b* then the same
statement also holds for *n* = *m* + 1
and filling the gaps in your "problem" text...
the statement is P(n) = "for every integer n greater than -3 , the number
2^n + 4 is greater than 33/8"
1. first step P(-3), statement hols when n = b with b = - 3
2^(-3) + 4 = 1/8 + 4 = 33/8 >= 0 statement is true
2. inductive step : Showing that if the statement holds for *n* = *m* ≥ *b*
then the same statement also holds for *n* = *m* + 1
given any integer number m greater or equal than b and 2^m + 4 >= 0 and
n = m+1
THEN
be 2^n + 4 = 2^(m+1) + 4 = 2^m 2 + 4 = (2^m + 4 - 4) 2 + 4 = (2^m + 4) 2
-4 2 + 4 = (2^m + 4) 2 - 4 >= 66/8 - 4 = 34/8 >= 33/8
if statement holds for m then it holds for n = m + 1
Because of 1/ and 2/ holds, then P(n) is true for every n greater or equal
than b.
-----------------------------------
Unfornulately for the inductive step, you need symbolic variables (n,m) to
solve some equation(s) or inequation(s)..and SAGE cannot do that when
forcing n to be integer.
Even the simple example :
n = var('n',domain='integer')
res = solve([n^2 == 3],n); print "res = ",res
returns the weird answer :
res = [
n == -sqrt(3),
n == sqrt(3)
]
because behind the scene, compute in the symbolic ring...without taking care of
the "integer"
constraint
...so sorry, I don't know.
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