The given expression is:

$\left( (n-1) \right)^2 - \frac{(n-2)(n-1)}{2}$

Let's simplify it step by step.

Step 1: Expand the square term $\left( (n-1) \right)^2$

$(n-1)^2 = n^2 - 2n + 1$

Step 2: Simplify the second term $\frac{(n-2)(n-1)}{2}$

First, expand $(n-2)(n-1)$: $(n-2)(n-1) = n^2 - 3n + 2$ Now divide by 2: $\frac{(n^2 - 3n + 2)}{2} = \frac{n^2}{2} - \frac{3n}{2} + 1$

Step 3: Combine both terms

Now subtract the second term from the first: $n^2 - 2n + 1 - \left( \frac{n^2}{2} - \frac{3n}{2} + 1 \right)$

Distribute the negative sign: $n^2 - 2n + 1 - \frac{n^2}{2} + \frac{3n}{2} - 1$

Step 4: Combine like terms

Combine the $n^2$ terms: $n^2 - \frac{n^2}{2} = \frac{2n^2}{2} - \frac{n^2}{2} = \frac{n^2}{2}$

Combine the $n$ terms: $-2n + \frac{3n}{2} = \frac{-4n}{2} + \frac{3n}{2} = \frac{-n}{2}$

Combine the constants: $1 - 1 = 0$

Final Expression:

The simplified result is: $\frac{n^2}{2} - \frac{n}{2}$

Alternatively, this can be factored as: $\frac{n(n-1)}{2}$

Would you like more details or have any questions? Here are some related questions to explore further:

1. How do you expand expressions like $(n-1)^2$?
2. What is the process of factoring quadratic expressions?
3. How do you handle fractions in algebraic expressions?
4. What are common techniques for simplifying complex expressions?
5. Can you explain how to combine like terms in more detail?

Tip: Always distribute negative signs carefully when simplifying expressions!