In an arithmetic progression (AP), the general formula for the $n$-th term is:

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

Where:

• $T_n$ is the $n$-th term,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

We are given:

• The 13th term ($T_{13}$) is 25, so: $T_{13} = a + 12d = 25$
• The 5th term ($T_5$) is also 25, so: $T_5 = a + 4d = 25$

We now have a system of two equations:

1. $a + 12d = 25$
2. $a + 4d = 25$

Step 1: Subtract the second equation from the first:

$(a + 12d) - (a + 4d) = 25 - 25$ $8d = 0$ $d = 0$

Step 2: Substitute $d = 0$ into one of the original equations:

Using $a + 4d = 25$: $a + 4(0) = 25$ $a = 25$

Step 3: Find the 7th term ($T_7$):

Using the formula $T_n = a + (n-1)d$ for the 7th term: $T_7 = a + 6d = 25 + 6(0) = 25$

Thus, the 7th term is 25.

Would you like more details or have any questions?

Here are some related questions:

1. How do you determine the common difference in an AP?
2. What happens when the common difference is zero?
3. Can an AP have all its terms the same? Why?
4. How can the sum of an arithmetic progression be calculated?
5. What changes in the formula for a geometric progression?

Tip: When the common difference in an arithmetic progression is zero, all the terms of the sequence will be the same.