In an arithmetic progression, the nth term is 6 less than the 7th term.
The common difference is -3.
Find the value of n.
n=

Let's start by using the formula for the \( n \)th term of an arithmetic progression (AP):

\[
a_n = a_1 + (n-1) \cdot d
\]

where:
- \( a_n \) is the \( n \)th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference.

We're given:
- The common difference \( d = -3 \).
- The \( n \)th term \( a_n \) is 6 less than the 7th term \( a_7 \).

So, we can write the relationship between the \( n \)th term and the 7th term as:

\[
a_n = a_7 - 6
\]

Now, let's express the 7th term using the formula:

\[
a_7 = a_1 + 6 \cdot d
\]

Substituting the value of \( d \) into the equation:

\[
a_7 = a_1 + 6 \cdot (-3) = a_1 - 18
\]

So, the \( n \)th term can be written as:

\[
a_n = (a_1 - 18) - 6 = a_1 - 24
\]

We also know that:

\[
a_n = a_1 + (n-1) \cdot (-3) = a_1 - 3(n-1)
\]

Setting the two expressions for \( a_n \) equal:

\[
a_1 - 3(n-1) = a_1 - 24
\]

Now, subtract \( a_1 \) from both sides:

\[
-3(n-1) = -24
\]

Divide by -3:

\[
n-1 = 8
\]

Finally, add 1 to both sides to find \( n \):

\[
n = 9
\]

So, the value of \( n \) is \( \boxed{9} \).

Would you like any further details on this solution?

Here are some related questions to explore:

1. How would the value of \( n \) change if the common difference were positive?
2. What is the first term (\( a_1 \)) of the arithmetic progression if \( n = 9 \)?
3. Can you derive a general formula to find \( n \) for any similar problem?
4. How does the value of \( a_7 \) influence the value of \( n \)?
5. What would happen if the difference between the \( n \)th term and the 7th term were greater than 6?
6. What is the sum of the first 9 terms of this arithmetic progression?
7. How do the positions of terms in an arithmetic progression affect their values?
8. How would you graph this arithmetic progression?

**Tip:** In arithmetic progressions, understanding the relationship between the terms and their positions can help solve more complex problems efficiently.

Learn how to find the value of n in an arithmetic progression problem where the nth term is 6 less than the 7th term and the common difference is -3. Step-by-step solution provided.

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