# 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^+#?

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(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

*(by the inductive hypothesis)*

#<= t_(k+1)" "# (by the previous proof)

#= (3n+8)/(n+3)#

# < (3n+9)/(n+3)#

#=3#

We have supposed true for