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The problem asks you to prove by induction that for all integers \( n \geq 1 \):

\[
1 \times 1! + 2 \times 2! + 3 \times 3! + \dots + n \times n! = (n+1)! - 1
\]

### **Steps for Proof by Mathematical Induction:**

#### **Base Case:**
- **For \( n = 1 \):**

\[
1 \times 1! = 1 = (1+1)! - 1 = 2! - 1 = 2 - 1 = 1
\]

So, the base case holds.

#### **Inductive Step:**
- **Assume the formula holds for some \( n = k \geq 1 \):**

\[
1 \times 1! + 2 \times 2! + 3 \times 3! + \dots + k \times k! = (k+1)! - 1
\]

- **We need to prove it holds for \( n = k+1 \):**

\[
1 \times 1! + 2 \times 2! + 3 \times 3! + \dots + k \times k! + (k+1) \times (k+1)!
\]

Using the inductive hypothesis:

\[
= \left[(k+1)! - 1\right] + (k+1) \times (k+1)!
\]

Factor \( (k+1)! \) from the expression:

\[
= (k+1)! \times [1 + (k+1)] - 1
\]

\[
= (k+1)! \times (k+2) - 1
\]

\[
= (k+2)! - 1
\]

This shows the formula holds for \( n = k+1 \) if it holds for \( n = k \).

#### **Conclusion:**
By the principle of mathematical induction, the formula 

\[
1 \times 1! + 2 \times 2! + 3 \times 3! + \dots + n \times n! = (n+1)! - 1
\]

is true for all \( n \geq 1 \).

Would you like any more details or explanations?

### Related Questions:

1. What is the significance of factorials in combinatorial problems?
2. How can induction be used to prove other summation formulas?
3. What are some common pitfalls in applying mathematical induction?
4. Can this method be adapted for non-integer sequences or sums?
5. How does this formula connect with other well-known factorial identities?

### Tip:
When working with mathematical induction, always carefully verify the base case and ensure your inductive step covers all possible cases.

Prove by induction that for all integers n >= 1: 1 x 1! + 2 x 2! + 3 x 3! + ... + n x n! = (n+1)! - 1.

Learn how to prove by induction that the sum 1 x 1! + 2 x 2! + 3 x 3! + ... + n x n! equals (n+1)! - 1. This step-by-step solution covers key concepts in mathematical induction and factorials, suitable for high school students in Grades 10-12.

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