To prove any result through mathematical induction we have to follow three steps. If all the steps are satisfied we can say that it is proved through mathematical induction. 

PMI and Factorials

PMI (Plus, Minus, Interesting) is a tool for critical thinking that helps to explore a statement or idea from different perspectives. In this case, we will use PMI to analyze the equation:

1.1! 2.2! 3.3! ___ n.n! = (n 1)! -1

Plus

The equation contains a pattern of factorials, which are mathematical functions that multiply a sequence of numbers.

Minus

The equation has a blank space that needs to be filled with a number that follows the pattern established by the sequence 1.1!, 2.2!, 3.3!, and n.n!.

Interesting

The equation relates the sum of a sequence of factorials to the difference between two factorials.

Solution

To solve the equation, we need to understand the pattern of the factorials in the sequence.

1.1! = 1
2.2! = 2 x 2 = 4
3.3! = 3 x 2 x 1 x 3 x 2 x 1 = 36

We notice that each factorial in the sequence is the product of a number and its corresponding decimal. For instance, 3.3! is the product of 3 x 2 x 1 x 0.3 x 0.2 x 0.1.

Therefore, the blank space in the equation should be filled with:

4.4! = 4 x 4 x 0.4 x 0.3 x 0.2 x 0.1 = 6.144

Now we can substitute the numbers in the equation and simplify it:

1! + 4! + 36 + 6.144 + n.n! = (n+1)! -1

We can simplify further by subtracting 1! from both sides:

4! + 36 + 6.144 + n.n! = (n+1)! -2

Finally, we can combine the terms on the left side and simplify the right side by factoring out (n+1):

40.144 + n.n! = (n+1)(n!) -2

Now we can solve for n by isolating the n! term and factoring out (n+1):

n.n! - (n+1)(n!) = -42.144

n!(n - (n+1)) = -42.144

n!(-1) = -42.144

n! = 42.144

However, this is not a valid solution because factorials are only defined for non-negative integers. Therefore, the equation has no solution.

Conclusion

Using PMI, we analyzed the equation 1.1! 2.2! 3.3! ___ n.n! = (n 1)! -1 and found that it had a pattern of factorials, a blank space that needed to be filled, and an interesting relationship between the sum of the sequence and the difference between two factorials. We solved the equation by understanding the pattern of the factorials and filling the blank space with 4.4!. We simplified the equation and found that