Prove that adding a number to itself n times results in the number squared | Step-by-Step Solution

Let's Explore This Step-by-Step! 🧮

What We're Solving: We need to examine whether adding a number n to itself n times actually equals n². This is a common misconception that we should investigate carefully!

The Approach: We're going to use the fundamental principle of multiplication as repeated addition. When we add the same number multiple times, we can represent this as multiplication. Let's work through this systematically to see what we actually get!

Step-by-Step Solution:

Step 1: Understand what "adding n to itself n times" means

• If we add n to itself n times, we have: n + n + n + ... + n (with n copies of the number n)
• This is the definition of multiplication: n × n

Step 2: Apply the multiplication rule

• n + n + n + ... + n (n times) = n × n = n²

Step 3: Example with n = 3

• Adding 3 to itself 3 times: 3 + 3 + 3 = 9
• And 3² = 9 ✓

Step 4: Another example

• Let's try n = 4
• Adding 4 to itself 4 times: 4 + 4 + 4 + 4 = 16
• And 4² = 16 ✓

Step 5: The general proof

• When we add n to itself n times, we're doing: n × n = n²
• This works because multiplication IS repeated addition!

The Answer: The statement is TRUE! Adding n to itself n times does indeed equal n². The mathematical expression is: n + n + n + ... + n (n terms) = n × n = n²

Memory Tip: Remember "Repeated Addition = Multiplication"! When you see the same number being added multiple times, count how many times it appears - that becomes your multiplication problem. So n added n times becomes n × n = n². 🌟

Great job working through this concept! Understanding WHY multiplication works as repeated addition is a fundamental building block in algebra.