 I'm Zor. Welcome to Unizor education. Today's topic is a problem for mathematical induction. This is problem number two. Okay, so here is the problem. Before we were talking about certain algebraic formulas, like left part is equal to right part, for instance, sum from 1 to n is equal to whatever the expression was on the right, and we were trying to prove this equality using the method of mathematical induction. Not only these equalities can prove it, this problem will be about inequality. I would like to prove an inequality of the following type, that n factorial is greater than 2 to the power of n. Now, by the way, n factorial is 1 times 2 times 3, etc. times n. And obviously, 2 to the power of n is 2 times 2 times 2. n times, and this is also n numbers from 1 to n. So the multiplication of n numbers from 1 to n is greater than the multiplication of numbers of n times repeated number 2. Okay, so this is the problem, and I would like you to prove it using the method of mathematical induction. By the way, let me just give you a little hint. If you will start checking this formula for n is equal to 1, or even 2, or even 3, it will be wrong. But basically, you feel that the more numbers on the left, the bigger the number n is, we are multiplying by a bigger and bigger number. So this thing is growing quite fast. This thing is growing with the same speed. It's always times 2. So eventually, this number should be greater than this number. We all feel this type. However, for initial number 1, 2, or even 3, this is not true. So the hint is that the first step in the mathematical induction, which is checking the formula for a certain specific initial value, this initial value should not be 1 in this case. It should be greater than 1. Actually, it's 4 or 5 or something. Because this formula which we would like to prove is supposed to be valid for big number n's. Any big number bigger than something. So this is the problem. Prove this formula for n greater or equal than 4. So 4 becomes your initial value. Perfect spot to pause the video and think about this yourself. Try to prove this inequality using the method of mathematical induction. And then you can click the play button again and I will continue with solution to this problem. So mathematical induction, three steps. Step number 1, check. For n equals to 4. Okay, for n equals 4, n factorial equals 1 times 2 times 3 times 4 equals to 24. 2 times n equals 2, 2 to the 4th degree, which is 16. Obviously, 24 is greater than 16. So the formula is true for n equals to 4. Check. Step number 2, assume for n equals to 5. Okay, so we assume that k factorial is greater than 2 to the kth degree. 3. Let n equals to k plus 1. We have to prove that the formula is true for n is equal to k plus 1. On the left, we will have k plus 1 factorial. Formula with n equals to k plus 1 is k plus 1 factorial. What is k plus 1 factorial? It's 1 times 2 times etc. times k and times k plus 1. Right? That's what k plus 1 factorial is by definition. Now, notice that these multiply together give you k factorial. So k plus 1 factorial is actually a k factorial times k plus 1. Now, we can use our assumption. This is our assumption. The k factorial is greater than 2 to the kth. Greater than, so instead of k factorial, we put 2 to the power of k times k plus 1. Now, k is at least greater than 4, right? So k plus 1 is definitely greater than 2. So it's greater than 2 to the kth times 2, which is equal to 2 to the power of k. Basically, we have, sorry, k plus 1. So that's what we have proved that the k plus 1 factorial is greater equal equal equal greater and greater again than 2 to the k plus 1. That's it. So we checked, we assumed for n is equal to k and then we proved that for n equal to k plus 1, this formula, this inequality is valid. And now, again, we don't write it down, but we'll always say it or at least we think about this, that considering we checked that the formula is true for n equals to 4 and considering that we can step always from k to k plus 1, therefore, we can step from n is equal to 4 to n is equal to 5, because our assumption is based on n is equal to 4 and from this assumption, we proved that it's true for k plus 1, so therefore, we proved for n equal to 5. From 5 to 6, from 6 to 7, etc., so the formula is true for n greater than 4 up to infinity. Well, that concludes this problem number 2. Thank you.