Prove That $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$

question 38

Short Answer

Prove that $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$ for all $n \geq 1$

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Definitions:

Free

Not under the control or in the power of another; able to act or be done as one wishes.

Enslaved

Made to work without pay and under threat of violence, deprived of freedom and rights; historically, referring primarily to African Americans in pre-Civil War United States.

First Great Awakening

A series of Christian revivals that swept Britain and its Thirteen Colonies in the Americas during the 1730s and 1740s, leaving a permanent impact on American Protestantism.

Evangelical Denominations

Groups within Christianity characterized by a strong emphasis on the authority of the Bible, the significance of Jesus' death and resurrection, and the imperative of evangelism and conversion.