[tex]H(n)=\frac{5}{n+1}[/tex]
Averate rate of change:
[tex]\begin{gathered} \lbrack a,b\rbrack \\ \\ ARC=\frac{H(b)-H(a)}{b-a} \end{gathered}[/tex]
For the given interval:
[tex]\begin{gathered} \text{ARC}_{\lbrack2,10\rbrack}=\frac{H(10)-H(2)}{10-2} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=\frac{\frac{5}{10+1}-\frac{5}{2+1}}{8} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=\frac{\frac{5}{11}-\frac{5}{3}}{8} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=\frac{\frac{15-55}{33}}{8} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=\frac{-\frac{40}{33}}{8} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=\frac{-40}{33\cdot8} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=-\frac{40}{264} \\ \\ \text{ARC}_{\lbrack2,10\rbrack}=-\frac{5}{33}=-0.\bar{15} \end{gathered}[/tex]
Then, the averate rate of change of the given function in the interval {2,10} is -5/33 (or -0.15)
