- #1
Woolyabyss
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Homework Statement
A sequence (an) of real numbers is defined by a(1) = 6, a(n) = (1/4)(2*a(n−1) − 3) for n >= 2. where n is a natural number
show the sequence is monotonically decreasing
Homework Equations
None
The Attempt at a Solution
I tried to prove it by induction.
Let P(n) be the predicate that a(n) < a(n-1)[/B]
base case
p(2)...
a(1) = 6
a(2) = (1/4)*(2*6-3) = 9/4
a(2) < a(1) ... base case is true
induction step
assume p(k) to be true
a(k) < a(k-1)
now we try to prove p(k+1) is true using or assumption about p(k)
a(k+1) = (1/4)*(2(k) - 3)
rearranging
a(k) = 2*a(k+1) - 3
using our assumption
a(k-1) < 2*a(k+1) - 3
I'm not sure where to go from here I can't think of anyway of relating a(k) and a(k+1) with an inequality.
any help would be appreciated.