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When simplifying algebraic expressions, it's crucial to identify 'like terms'.
Like terms are terms that have exactly the same variables raised to the same powers.
In this expression, \(a_{1} + a_{n}\), we notice that each parenthesis contains the same terms.
This shows that we are dealing with like terms.

When you see that the terms inside the parentheses are the same, you can treat them as a single unit.
This is a fundamental step because it simplifies the process of combining the terms.

Remember that you cannot combine terms that are not alike. For instance, \(a_1 \) and \(a_2\) are not 'like terms' because they have different subscripts.
This step sets the foundation for using the distributive property in the next step.

The distributive property is a rule that lets you multiply a sum by multiplying each addend separately and then adding the results.
The formula for the distributive property is \(a(b + c) = ab + ac\).
In our exercise, we apply the distributive property to simplify the expression:

Given the expression \left(a_{1}+a_{n}\right) + \left(a_{1}+a_{n}\right) + \left(a_{1}+a_{n}\right)\:
- We have three like terms.
- We can see that \left(a_{1}+a_{n}\right)\ is common in all three terms.

Using the distributive property, we factor out \(a_{1}+a_{n}\) and multiply it by 3:

\[3 \times \left(a_{1}+a_{n}\right)\]

Now the expression is simplified by using the distributive property correctly, streamlining the calculation.

Factoring is a technique to simplify an expression by finding a common factor in each term.
It works by writing the original expression as a product of its factors.
In this exercise, we noticed that \(a_{1} + a_{n}\) is a common factor in all terms.

Factoring out the common term is similar to the distributive property but in reverse.
In our case, after identifying the common term, we factored it out and simplified the expression to:

\[3 \times \left(a_{1}+a_{n}\right)\]

This step-by-step approach of factoring out like terms helps in reducing complex algebraic expressions into simpler forms.
Always look for common factors as it makes the simplification process much easier and manageable.