Find the sum of all the times in the progression 180, 175, 170,…, 25.

The sequence provided is an arithmetic progression (AP) where:

- The first term \( a = 180 \),
- The common difference \( d = -5 \) (since each term decreases by 5), and
- The last term \( l = 25 \).

To find the sum of all terms in the progression, we first need to determine the number of terms, \( n \), using the formula for the \(n\)-th term of an arithmetic progression:

\[
a_n = a + (n-1) \cdot d
\]

We know the last term \( a_n = 25 \), so:

\[
25 = 180 + (n-1) \cdot (-5)
\]

Simplifying:

\[
25 = 180 - 5(n-1)
\]

\[
25 - 180 = -5(n-1)
\]

\[
-155 = -5(n-1)
\]

\[
n-1 = 31
\]

\[
n = 32
\]

Thus, there are 32 terms in the sequence.

Next, we use the sum formula for an arithmetic progression:

\[
S_n = \frac{n}{2} \cdot (a + l)
\]

Substituting the known values:

\[
S_{32} = \frac{32}{2} \cdot (180 + 25)
\]

\[
S_{32} = 16 \cdot 205 = 3280
\]

Therefore, the sum of all the terms in the progression is \( \boxed{3280} \).

Would you like more details or have any questions?

Here are some related questions you might explore:
1. How do you derive the formula for the sum of an arithmetic progression?
2. What would be the sum if the common difference was positive instead of negative?
3. Can you find the sum of the first 20 terms in the sequence 100, 95, 90, ..., using a similar approach?
4. What happens if the sequence is extended to include negative values? What would the sum be?
5. How would you calculate the sum if the progression followed a geometric sequence instead?

**Tip**: Always identify the common difference \(d\) and the first term \(a\) before using any AP formula!

Find the sum of all the terms in the progression 180, 175, 170,…, 25.

Learn how to find the sum of an arithmetic progression with the first term 180, a common difference of -5, and the last term 25. This step-by-step solution includes calculating the number of terms and applying the sum formula for an arithmetic sequence. Suitable for students in Grades 9-11.

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