Animated Solution for Mathematics - Binomial Theorem: Use mathematical induction to show that $(25{)}^{n+1}-24n+5735$ is divisible by $(24{)}^2$ for all $n=1,2,.......$

Mastering Induction

Welcome, future engineers! Today, we are going to demystify one of the most elegant tools in your mathematical arsenal: Mathematical Induction.

Imagine a long line of dominoes stretching to infinity. To prove that every single one will fall, you only need to do two things: push the first one over, and ensure that if any domino falls, it knocks over the next one. That is exactly what we are doing with this divisibility problem.

The Base Case

We are tasked with proving that $P(n):(25{)}^{n+1}-24n+5735$ is divisible by $24^2$, which is $576$, for all $n\in \mathbb{N}$.

Our first step is the base case, $n=1$. We substitute $n=1$ into our expression:

This simplifies to $25^2-24+5735$, which is $625-24+5735=6336$.

Now, we check if $6336$ is divisible by $576$. A quick division, confirms it! The first domino has fallen. The statement holds for $n=1$.

The Inductive Hypothesis

Now, we assume the statement is true for some arbitrary natural number $k$. This is our 'Inductive Hypothesis.'

We assume: for some integer $m$.

Here is the pro-tip: do not just stare at this equation. Manipulate it! We want to isolate the highest power term, $25^{k+1}$, because it is the most difficult part of the expression.

By rearranging, we get: Keep this in your back pocket; it is the key that will unlock the next step.

The Inductive Step

Now, we must prove that if $P(k)$ is true, then $P(k+1)$ must also be true. Let's write out $P(k+1)$ by replacing $n$ with $k+1$:

This simplifies to $25^{k+2}-24k-24+5735$. To connect this to our hypothesis, we split the exponential term:

Now, we substitute our isolated expression from the hypothesis:

The Victory Lap

Now, we expand and simplify. Distributing the $25$, we get:

This looks intimidating, but watch the magic happen. We have:

Combining the $k$ terms, $600k-24k=576k$. Combining the constants, $-143375+5711=-137664$.

Our expression is now:

We can factor out $576$ from the first two terms easily. For the constant, $137664÷576=239$.

Thus:

Since $m$ and $k$ are integers, the term in the bracket is an integer. We have proven that $P(k+1)$ is a multiple of $576$.

By the Principle of Mathematical Induction, the statement is true for all $n\in \mathbb{N}$. You have conquered the problem!