What is the value of n in geometric sequence if the given are An=-1024,A1=-4 and r=4

In a geometric sequence, the formula for the \( n \)-th term (\( A_n \)) is given by:

\[
A_n = A_1 \cdot r^{n-1}
\]

Where:
- \( A_n \) is the \( n \)-th term,
- \( A_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number we need to find.

We are given:
- \( A_n = -1024 \),
- \( A_1 = -4 \),
- \( r = 4 \).

Substitute the known values into the formula:

\[
-1024 = -4 \cdot 4^{n-1}
\]

To solve for \( n \), first divide both sides by \( -4 \):

\[
256 = 4^{n-1}
\]

Now, express 256 as a power of 4:

\[
256 = 4^4
\]

So, we have:

\[
4^{n-1} = 4^4
\]

Since the bases are the same, we can equate the exponents:

\[
n - 1 = 4
\]

Solve for \( n \):

\[
n = 5
\]

Thus, the value of \( n \) is \( \boxed{5} \).

---

Do you want more details or have any questions? Here are some related questions to consider:

1. How would the formula change if \( r \) were a fraction instead of a whole number?
2. What is the formula for the sum of the first \( n \) terms of a geometric sequence?
3. How would the solution be affected if \( A_n \) were a positive value instead?
4. What is the importance of the common ratio \( r \) in a geometric sequence?
5. How can you determine whether a sequence is geometric or arithmetic?

**Tip:** Always check whether the common ratio \( r \) is positive or negative, as it can significantly affect the sequence's behavior!

What is the value of n in a geometric sequence if the given are An=-1024, A1=-4, and r=4?

Find the term number n in a geometric sequence where An = -1024, A1 = -4, and r = 4. This problem explores geometric sequences, exponent rules, and step-by-step calculation methods. Ideal for students in Grades 8-10.

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