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The problem states the following:

We are given an arithmetic progression \( (a_n)_{n \geq 1} \) with the properties:

- \( a_7 - a_1 = 12 \)
- \( a_1 + a_2 = 4 \)

We are asked to determine the term \( a_{100} \).

### Solution:

An arithmetic progression has the form:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \( a_1 \) is the first term, and \( d \) is the common difference.

#### Step 1: Using the first property \( a_7 - a_1 = 12 \)

\[
a_7 = a_1 + 6d
\]
So,
\[
a_7 - a_1 = (a_1 + 6d) - a_1 = 6d
\]
From the problem, we know:
\[
6d = 12 \quad \Rightarrow \quad d = 2
\]

#### Step 2: Using the second property \( a_1 + a_2 = 4 \)

We know that:
\[
a_2 = a_1 + d
\]
Thus:
\[
a_1 + a_2 = a_1 + (a_1 + d) = 2a_1 + d
\]
Substituting \( d = 2 \):
\[
2a_1 + 2 = 4 \quad \Rightarrow \quad 2a_1 = 2 \quad \Rightarrow \quad a_1 = 1
\]

#### Step 3: Finding \( a_{100} \)

Now that we know \( a_1 = 1 \) and \( d = 2 \), we can find \( a_{100} \):
\[
a_{100} = a_1 + (100 - 1) \cdot d = 1 + 99 \cdot 2 = 1 + 198 = 199
\]

### Final Answer:
\[
a_{100} = 199
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. How do you derive the general term of an arithmetic progression?
2. Can you explain how the common difference \( d \) affects the progression?
3. How do you solve similar problems if the arithmetic progression starts from a different term, like \( a_3 \)?
4. What happens if we are given the sum of terms instead of their individual values?
5. Can arithmetic progressions have negative common differences? What does that imply?

**Tip:** Always check if the provided information allows you to find both the first term and the common difference—this is key to solving arithmetic progression problems.

Se consideră progresia aritmetică (a_n)_{n ≥ 1} cu proprietatea că a_7 - a_1 = 12 și a_1 + a_2 = 4. Determinați termenul a_{100}.

Learn how to solve an arithmetic progression problem where a_7 - a_1 = 12 and a_1 + a_2 = 4. Find the 100th term using key algebraic concepts and properties of arithmetic progressions. Suitable for students in Grades 9-10.

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