•(〖n-1)〗^2- ((n-2)(n-1))/2 =(n-1)n/2

We are given the equation:

\[
(n-1)^2 - \frac{(n-2)(n-1)}{2} = \frac{(n-1)n}{2}
\]

Let's solve it step by step.

### Step 1: Expand the terms
Start by expanding each term on the left-hand side.

1. Expand \((n-1)^2\):

\[
(n-1)^2 = n^2 - 2n + 1
\]

2. Expand \(\frac{(n-2)(n-1)}{2}\):

\[
(n-2)(n-1) = n^2 - 3n + 2
\]
Thus:
\[
\frac{(n^2 - 3n + 2)}{2} = \frac{n^2}{2} - \frac{3n}{2} + 1
\]

### Step 2: Substitute the expanded terms
Substitute both expansions into the original equation:

\[
(n^2 - 2n + 1) - \left( \frac{n^2}{2} - \frac{3n}{2} + 1 \right) = \frac{(n-1)n}{2}
\]

### Step 3: Simplify the left-hand side
Now, combine like terms on the left-hand side:

\[
n^2 - 2n + 1 - \left( \frac{n^2}{2} - \frac{3n}{2} + 1 \right)
\]

Distribute the negative sign:

\[
n^2 - 2n + 1 - \frac{n^2}{2} + \frac{3n}{2} - 1
\]

Now, combine like terms:

\[
n^2 - \frac{n^2}{2} = \frac{n^2}{2}, \quad -2n + \frac{3n}{2} = -\frac{n}{2}, \quad 1 - 1 = 0
\]

So the left-hand side simplifies to:

\[
\frac{n^2}{2} - \frac{n}{2}
\]

### Step 4: Compare both sides
Now, the equation becomes:

\[
\frac{n^2}{2} - \frac{n}{2} = \frac{(n-1)n}{2}
\]

Expand the right-hand side:

\[
\frac{(n-1)n}{2} = \frac{n^2 - n}{2}
\]

Thus, we have:

\[
\frac{n^2}{2} - \frac{n}{2} = \frac{n^2}{2} - \frac{n}{2}
\]

Both sides are equal, so the equation is true.

### Conclusion:
The equation holds true for all values of \( n \).

---

Would you like further clarification on any part of the solution?

Here are five related questions to explore:
1. How can you solve quadratic equations with fractions?
2. How would you approach simplifying more complex algebraic fractions?
3. What is the process for expanding binomials like \((n-1)^2\)?
4. What happens when you encounter an equation that seems unsolvable? How do you analyze it?
5. How does solving linear equations compare to solving quadratic ones?

**Tip:** Always expand and simplify carefully when working with equations involving squares and fractions to avoid small errors.

The query submitted by the user: (〖n-1)〗^2 - ((n-2)(n-1))/2 = (n-1)n/2

Explore how to simplify the equation (n-1)^2 - ((n-2)(n-1))/2 = (n-1)n/2 step-by-step. This solution delves into expanding binomials, combining like terms, and handling fractional equations. Ideal for students in Grades 9-12.

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