Animated Solution for Mathematics - Basic Mathematics: Given $n^4<10^n$ for a fixed positive integer $n\ge 2$, prove that $(n+1{)}^4<10^{n+1}$.

Imagine you are standing at the base of a mountain, looking at two paths. One path is defined by a polynomial, $n^4$, and the other by an exponential, $10^n$.

We are told that for any integer $n\ge 2$, the polynomial path is always lower than the exponential path. Our mission is to prove that this relationship holds for the next step, $n+1$.

This is the essence of mathematical induction, but we will solve it with a more intuitive tool: the ratio strategy.

Decoding the Growth

When we want to compare two terms, say $A_{n+1}$ and $A_n$, the most insightful question we can ask is: "How much larger is the next term compared to the current one?"

We define this growth factor as the ratio $R$:

By analyzing this ratio, we can determine if the polynomial is growing faster or slower than the exponential base. If $R<10$, then the polynomial is not growing fast enough to catch up to the exponential, and the inequality $n^4<10^n$ will hold for the next term.

Let us simplify our ratio. We have:

This is a beautiful transformation. We have moved from a complex fraction to a simple expression involving $\frac{1}{n}$.

Now, we apply our constraint: $n\ge 2$. As $n$ grows, $\frac{1}{n}$ shrinks. Therefore, the expression ${\left(1+\frac{1}{n}\right)}^4$ is at its largest when $n$ is at its smallest.

For $n=2$, we have:

Calculating $\frac{81}{16}$ gives us $5.0625$. This is the crucial moment of the proof.

We have shown that the growth factor of our polynomial is at most $5.0625$. Since $5.0625<10$, we have proven that the polynomial's growth is strictly bounded by the exponential base.

Mathematically, this means:

Now, we invoke our given condition: $n^4<10^n$. Substituting this into our inequality, we get:

By the laws of exponents, $10\cdot 10^n=10^{n+1}$. Thus:

We have successfully navigated the path and reached the summit. This is the power of mathematical reasoning: taking a complex problem, breaking it down into a ratio, bounding the growth, and synthesizing the final result.