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

question 255

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 $b _ { n }$ .

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Definitions:

Cash Flows

The total amount of money being transferred into and out of a business, often used as an indicator of a company's financial strength.

Discount Rate

The discount rate employed in discounted cash flow analysis to ascertain the present worth of future cash flows.

Annual Salary

The total amount of money that an employee earns in a year from their employer for their services.

Starting Bonus

A preliminary financial incentive given to a new employee as an enticement to join a company.