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 all#n 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