Question

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

Answer 1

To convert the recursive form \( a_1 = 5 \) and \( a_n = 2 a_{n-1} \) into an explicit form, we can calculate the first few terms:

- \( a_1 = 5 \)
- \( a_2 = 2 a_1 = 2 \cdot 5 = 10 \)
- \( a_3 = 2 a_2 = 2 \cdot 10 = 20 \)
- \( a_4 = 2 a_3 = 2 \cdot 20 = 40 \)

It appears that the terms are multiplying by 2 each time. Let's notice the pattern:

- \( a_1 = 5 \)
- \( a_2 = 5 \cdot 2^1 \)
- \( a_3 = 5 \cdot 2^2 \)
- \( a_4 = 5 \cdot 2^3 \)

From this, we can conclude that in general:

\[
a_n = 5 \cdot 2^{n-1}
\]

So the explicit form of the sequence is:

\[
a_n = 5 \cdot 2^{n-1}
\]

Therefore, the correct response is:

\[
a_n = 5(2)^{n-1}
\]