Use the Principle of Mathematical Induction to Prove That$$4 \mid \left( 9 ^ { n } - 5 ^ { n } \right) \text { for all } n \geq 0$$

Question

Examlex · Accepted Answer

The Answer is

P ( 0 ) : 4 \mid 1 - 1

is true since

4 \mid 0 . \quad P ( k ) \rightarrow P ( k + 1 ) : 9 ^ { k + 1 } - 5 ^ { k + 1 } = 9 \left( 9 ^ { k } - 5 ^ { k } \right) + 5 ^ { k } ( 9 - 5 )

. Each term is divisible by

4 : 4 \rfloor 9 ^ { k } - 5 ^ { k } (

bv

P ( k ) )

and 4

\rfloor 9 - 5

,

Use the Principle of Mathematical Induction to Prove That $4 \mid \left( 9 ^ { n } - 5 ^ { n } \right) \text { for all } n \geq 0$