Use Mathematical Induction to Prove the Property for All Positive$n$

Question

Examlex · Accepted Answer

The Answer is 1)

n = 1

:

\begin{array} { r } { \left[ a ^ { 1 } ight] ^ { 4 } \stackrel { ? } { = } a ^ { 4 \cdot 1 } } \{ [ a ] ^ { 4 } \stackrel { ? } { = } a ^ { 4 } } \a ^ { 4 } = a ^ { 4 }\end{array}

The statement is true for

n = 1

.
2) Assume

\left[ a ^ { k } ight] ^ { 4 } = a ^ { 4 k }

. Then,

\begin{aligned}( a ) ^ { 4 } \left[ a ^ { k } ight] ^ { 4 } & = ( a ) ^ { 4 } \left( a ^ { 4 k } ight) \{ \left[ a \cdot a ^ { k } ight] ^ { 4 } } & = a ^ { 4 + 4 k } \{ \left[ a ^ { k + 1 } ight] ^ { 4 } } & = a ^ { 4 ( k + 1 ) }\end{aligned}

By mathematical induction, the property is true for all positive values of

n

.

Use Mathematical Induction to Prove the Property for All Positive $n$