Question

How to prove the following inequality by induction.

show that 7*n^3 <5^n for all n>4 (natural numbers)

Answer 1

So I proved the base case for n=4 \[7*n^3<5^n => 448 <625\]

then I try to check for n +1 but get stuck so I know that
\[ \text{for } n+1 we have 7(n+1)^3<5^{n+1} \]
\[ 5*7n^3\le 7(n+1)^3 <5*5n^3 \] but then I am stuck to show that this is true.

Answer 2

\[7(n+1)^3=7n^3+3\cdot 7n^2+3\cdot 7n+1\]

\[<7n^3+n\cdot 7n^2+n\cdot 7n+1\]
\[=7n^3+7n^3+ 7n^2+1\]
\[<7n^3+7n^3+ 7n^3+7n^3\]
\[<5^n+5^n+ 5^n+5^n=4\cdot 5^n>5\cdot5^n=5^{n+1}\]

Answer 3

last inequality should be \(<\)

Answer 4

oh wow thank you ! @Zarkon

Answer 5

np