A Sequence of Right Triangles $A _ { 1 }$ $A _ { 2 }$

Essay

A sequence of right triangles, $A _ { 1 }$ , $A _ { 2 }$ , $A _ { 3 }$ , ... is given in the figure below: (a) Let $a _ { n } = \operatorname { area } \left( A _ { n } \right)$ . Determine an expression for $a _ { n }$ and find the limit of $a _ { n }$ .(b) Let $b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$ . Use geometric reasoning to determine the limit of $b _ { n }$ .