# Verify using Mathematical Induction

#### Guilmon

##### New member
Verify that

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

is true using induction

This problem has me stumped...

#### MarkFL

Staff member
Re: Mathematical Induction

Hello and welcome to MHB, Guilmon! Have you verified this is true for the base case $P_1$?

#### Prove It

##### Well-known member
MHB Math Helper
Re: Mathematical Induction

Verify that

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

is true using induction

This problem has me stumped...
If you theoretically could line up a row that is infinitely long, then you know that any domino will push the next one over, as long as the first domino is pushed. Mathematical induction works the same way. You need to prove a statement for a base case (which is equivalent to pushing the first domino), and then you need an inductive step, which is to prove that IF the statement is true for an arbitrary case, THEN the statement will be true for the next (which is equivalent to any domino pushing the next one over).

So how do you think you would go about proving the base case?

#### chisigma

##### Well-known member
Re: Mathematical Induction

Verify that

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

is true using induction

This problem has me stumped...
With simple steps You can verify that is...

$(n+1)!-1 = (n+1)\ n! -1 = n\ n! + n! -1$ (1)

What does You suggest (1)?...

Kind regards

$\chi$ $\sigma$

#### Guilmon

##### New member
The nth case I understand. It is the n+1th case that has me stumped.

#### MarkFL

Staff member
After having verified $P_1$ is true, I would next state the induction hypothesis $P_k$:

$$\displaystyle \sum_{i=1}^k(i\cdot i!)=(k+1)!-1$$

Now, our goal is to algebraically transform $P_k$ into $P_{k+1}$.

What do you think we should do to both sides of $P_k$ to meet that goal?

Last edited:

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
The nth case I understand. It is the n+1th case that has me stumped.
I am still not sure that you understand how induction works and what has to be proved (not how it is proved). There is no separate nth and (n+1)th cases in a proof by induction.

To make sure we are on the same page, I recommend starting from the very beginning: identifying the property P(n) that you need to prove for all n. Without this step, everything else is useless. Note that P(n) has to be true or false for each concrete n; in particular, P(n) cannot be a number such as 1(1!)+2(2!)+...+n(n!). Identifying P(n) in simple cases is easy: just remove the words "for all n" (if they are present) from the claim you are asked to prove. In this case, P(n) is the equality 1(1!)+2(2!)+...+n(n!) = (n+1)! - 1.

Next, you need to know how to write P(0) (or P(1)) and P(n+1). To do this, replace n with 0 (or 1, or n+1, respectively) everywhere in P(n).

Now you should be able to write the base case P(0) (or P(1)) and the induction step: "For all n, P(n) implies P(n+1)". Write both claims explicitly, replacing P by its definition. Only when you do this, you'll know what you need to prove. How to prove it is the following step.