$(1)如圖所示 $
$(2)42,n(n+1).提示:圖④中a_{6}=6×7=42,根據以上規(guī)律,正n邊形的“擴展圖形”中a_{n}=n(n+1) $
$(3)因為\frac{1}{a_{3}}=\frac{1}{3}-\frac{1}{4},\frac{1}{a_{4}}=\frac{1}{4}-\frac{1}{5},\frac{1}{a_{5}}=\frac{1}{5}-\frac{1}{6},\cdots,且\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}+\cdots+\frac{1}{a_{n}}=\frac{97}{300},所以\frac{1}{3}-\frac{1}{n+1}=\frac{97}{300},解得n=99\ $
