Use the Principle of Mathematical Induction to Find the Equivalent $1 + 2 \cdot 2 + 3 \cdot 2 ^ { 2 } + \ldots + n \cdot 2 ^ { n - 1 }$

question 8

Multiple Choice

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $1 + 2 \cdot 2 + 3 \cdot 2 ^ { 2 } + \ldots + n \cdot 2 ^ { n - 1 }$

• A) \(1 + ( n - 1 ) 2 ^ { n }\)
• B) \(( n - 1 ) 2 ^ { n }\)
• C) \(9 + ( 5 n - 9 ) 2 ^ { n }\)
• D) \(11 ^ { n } - 1\)
• E) \(( 5 n - 9 ) 2 ^ { n }\)

---

Definitions:

Club Meeting

An organized gathering of members of a club or organization, typically for discussing matters of common interest.

Attendance

The act of being present at a location or event, often tracked for purposes such as education, employment, or participation in activities.

Number

An entity employed in mathematics for counting, measuring, and naming purposes.

Persons

Human individuals or groups considered collectively, especially in legal or formal contexts.