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

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) $( 5 n - 9 ) 2 ^ { n }$