Write the Word or Phrase That Best Completes Each Statement $7 ^ { \mathrm { n } } - 2 ^ { \mathrm { n } }$

question 178

Essay

Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-Use the Principle of Mathematical Induction to show that the statement "5 is a factor of $7 ^ { \mathrm { n } } - 2 ^ { \mathrm { n } }$ " is true for all natural numbers. (Hint: $7 ^ { \mathrm { k } + 1 } - 2 ^ { \mathrm { k } + 1 } = 7 \left( 7 ^ { \mathrm { k } } - 2 ^ { \mathrm { k } } \right) + 5 \cdot 2 ^ { \mathrm { k } }$ )

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