Question

Convert the recursive form a1=5, an=2an−1 to the explicit form. (1 point) Responses an=5(2)n+1 A sub n is equal to 5 times 2 raised to the n plus 1 power an=5(2)n−1 A sub n is equal to 5 times 2 raised to the n minus 1 power an=3(7)n A sub n is equal to 3 times 7 to the n th power an=2(5)n−1

Answer 1

To convert the recursive form of the sequence \( a_1 = 5 \) and \( a_n = 2a_{n-1} \) into an explicit form, we can observe how the terms are generated.

1. **Calculate the first few terms:**
- \( a_1 = 5 \)
- \( a_2 = 2a_1 = 2(5) = 10 \)
- \( a_3 = 2a_2 = 2(10) = 20 \)
- \( a_4 = 2a_3 = 2(20) = 40 \)

2. **Identify a pattern:**
From the calculations, we observe:
\[
a_1 = 5 = 5 \times 2^0
\]
\[
a_2 = 10 = 5 \times 2^1
\]
\[
a_3 = 20 = 5 \times 2^2
\]
\[
a_4 = 40 = 5 \times 2^3
\]

3. **Generalize to find an explicit formula:**
From this pattern, we see that:
\[
a_n = 5 \times 2^{n-1}
\]

Thus, the explicit form of the sequence is:
\[
a_n = 5 \times 2^{n-1}
\]

Looking at the provided responses, the correct explicit form corresponds to:
\[
a_n = 5(2)^{n-1} \quad \text{(A sub n is equal to 5 times 2 raised to the n minus 1 power)}
\]

So, the correct answer is:
**an = 5(2)^(n−1)**.