To solve this problem, let's define the terms of the Arithmetic Progression (AP) and use the given information:

Let:

• $a$ be the first term of the AP
• $d$ be the common difference of the AP

The general formula for the $n$-th term of an AP is:

$T_n = a + (n - 1) \cdot d$

Step 1: Use the given terms

We are given:

• The 2nd term, $T_2 = 13$
• The 5th term, $T_5 = 25$

Using the formula for the $n$-th term: $T_2 = a + (2 - 1) \cdot d = a + d = 13 \quad \text{(Equation 1)}$ $T_5 = a + (5 - 1) \cdot d = a + 4d = 25 \quad \text{(Equation 2)}$

Step 2: Solve for $a$ and $d$

From Equation 1: $a + d = 13 \quad \text{(Equation 1)}$ From Equation 2: $a + 4d = 25 \quad \text{(Equation 2)}$

Now, subtract Equation 1 from Equation 2 to eliminate $a$: $(a + 4d) - (a + d) = 25 - 13$ $3d = 12$ $d = 4$

Substitute $d = 4$ back into Equation 1: $a + 4 = 13$ $a = 9$

Step 3: Find the 7th term

Now that we know $a = 9$ and $d = 4$, we can find the 7th term:

$T_7 = a + (7 - 1) \cdot d = 9 + 6 \cdot 4 = 9 + 24 = 33$

Final Answer:

The 7th term of the AP is $\boxed{33}$.

Would you like more details or have any further questions?

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Here are 5 related questions for further exploration:

1. What is the 10th term of the AP?
2. What is the sum of the first 7 terms of the AP?
3. How would the 3rd term of the AP be calculated?
4. Can you find the common difference if the 4th term is given?
5. What if the first term were unknown, but the 3rd and 7th terms were provided?

Tip: In an arithmetic progression, the common difference remains constant between consecutive terms. Use this to solve efficiently.