If a, start subscript, 1, end subscript, equals, 3*a*1​=3 and a, start subscript, n, end subscript, equals, 2, a, start subscript, n, minus, 1, end subscript*an*​=2*an*−1​ then find the value of a, start subscript, 6, end subscript*a*6​


Given the recursive sequence defined by:

\[
a_1 = 3
\]
\[
a_n = 2a_{n-1} \quad \text{for } n > 1
\]

We need to find the value of \( a_6 \).

Let's calculate each term up to \( a_6 \) step by step:

1. **Finding \( a_2 \):**

\[
a_2 = 2a_1 = 2 \times 3 = 6
\]

2. **Finding \( a_3 \):**

\[
a_3 = 2a_2 = 2 \times 6 = 12
\]

3. **Finding \( a_4 \):**

\[
a_4 = 2a_3 = 2 \times 12 = 24
\]

4. **Finding \( a_5 \):**

\[
a_5 = 2a_4 = 2 \times 24 = 48
\]

5. **Finding \( a_6 \):**

\[
a_6 = 2a_5 = 2 \times 48 = 96
\]

So, the value of \( a_6 \) is:

\[
\boxed{96}
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore:
1. What is the value of \( a_7 \) in this sequence?
2. Can you derive a general formula for \( a_n \) based on the given recursive relation?
3. How does the sequence change if the initial value \( a_1 \) is different?
4. What type of sequence is this, and what are its general properties?
5. How would the sequence be affected if the multiplication factor in the recursive relation was 3 instead of 2?

**Tip:** For geometric sequences, the nth term can often be expressed directly using the initial term and the common ratio raised to a power.

Explore the solution to a recursive sequence problem where a_1 = 3 and a_n = 2a_{n-1}. Calculate a_6 step-by-step and understand the recursive sequence pattern. Suitable for advanced high school students.

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