Can you use mathematical induction to prove that t_n >= t_(n-1) for all n in ZZ^+ for a sequence with the general term: t_n=(3n+5)/(n+2), n in ZZ^+?
(b) And hence, or otherwise, prove that 8/3 <= t_n <= 3 for all n in ZZ^+
(b) And hence, or otherwise, prove that
1 Answer
Induction does not seem to help prove the initial conjecture, but seems better suited for proving part (b).
Proof:
=(3n+5)/(n+2)-(3n+2)/(n+1)
=((3n+5)(n+1)-(3n+2)(n+2))/((n+1)(n+2))
=1/((n+1)(n+2)
>0 for alln in ZZ^+
(b)
Proof: (by induction)
Base case: For
Inductive hypothesis: Suppose that
Induction step: We wish to show that
<= t_(k+1)" " (by the previous proof)
= (3n+8)/(n+3)
< (3n+9)/(n+3)
=3
We have supposed true for