Use the Formula $1 + r + r ^ { 2 } + \cdots + r ^ { n } = \frac { r ^ { n + 1 } - 1 } { r - 1 }$

question 35

Essay

Use the formula $1 + r + r ^ { 2 } + \cdots + r ^ { n } = \frac { r ^ { n + 1 } - 1 } { r - 1 }$
(for all real numbers $r \neq 1$ and for all integers $n \geq 0$ ) to find
$$2 + 2 ^ { 2 } + 2 ^ { 3 } + \cdots + 2 ^ { m }$$
where m is an integer that is at least 1.

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Definitions:

Balanced Trade

A situation in which the value of a country's exports equals the value of its imports over a certain period.

Net Exports

The value of a country's exports minus its imports; a positive number indicates a trade surplus, while a negative one indicates a trade deficit.

Trade Surplus

A situation in which a country's exports exceed its imports.

Exports

Goods or services sent from one country to another for sale or trade.