There were multiple options for this question, but ultimately I am problems with just two of those ie.
$n^4<10^n$
$2^{3n} >7n+1$
Now following the basic induction process, for the first option I got $$k^4 <10^k$$
$$(k+1)^4<10^{k+1}$$ $$\frac{(k+1)^4}{10}<10^k$$
I am unable to establish a direct relation between the two inequalities, because the LHS is lesser than the common RHS.
The second is false - take $n = 1$. If it were true, we'd have $$2^3 = 8 \; > \; 7 \times 1 + 1 = 8 \implies 8 > 8$$ For the former, note that it's true for $n \in \lbrace 1, 2 \rbrace$. If we assume it's true for $n = k \geq 2$, then \begin{align*} (k+1)^4 &= k^4\left(1 + \frac{4}{k} + \frac{6}{k^2} + \frac{4}{k^3} + \frac{1}{k^4} \right) \\ &\leq 10^k \cdot \left( 1 + \frac{4}{2} + \frac{6}{2^2} + \frac{4}{2^3} + \frac{1}{2^4} \right) \\ &< 10^k \cdot \left( 1 + 2 + 2 + 1 + 1 \right) \\ &= 10^k \cdot 7 \\ &< 10^{k+1} \end{align*} where in obtaining the first inequality, we've used $k^4 < 10^k$ for the first term, and $k \geq 2$ inside the brackets.
