解：$(3)$示意图不唯一，如图所示
$ (4)(a_{1}+a_{2})^2=a_{1}^2+2a_{1}a_{2}+a_{2}^2，$共有$1+2=3($项)；
$(a_{1}+a_{2}+a_{3})^2=a_{1}^2+a_{2}^2+a_{3}^2+2a_{1}a_{2}+2a_{2}a_{3}+2a_{1}a_{3}$
共有$1+2+3=6($项)
∴将代数式$(a_{1}+a_{2}+a_{3}+···+a_{20})^2$
展开、合并同类项后共有$1+2+3+···+20=\frac {(1+20)×20}{2}=210($项)
