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=2(5)n−1
A sub n is equal to 2 times 5 raised to the n minus 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

Answer 1

The recursive relation is given as:

1. \( a_1 = 5 \)
2. \( a_n = 2a_{n-1} \) for \( n > 1 \)

To derive the explicit form, let's analyze the terms step by step:

- For \( n = 1 \):
\[
a_1 = 5
\]

- For \( n = 2 \):
\[
a_2 = 2a_1 = 2 \times 5 = 10
\]

- For \( n = 3 \):
\[
a_3 = 2a_2 = 2 \times 10 = 20
\]

- For \( n = 4 \):
\[
a_4 = 2a_3 = 2 \times 20 = 40
\]

From the above calculations, we can see a pattern forming. We can observe that:

\[
a_n = 2a_{n-1}
\]

This suggests each term is generated by multiplying the previous term by 2. We can express any term \( a_n \) as follows:

\[
a_n = 2^{n-1} a_1
\]

Substituting \( a_1 = 5 \):

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

Thus, the explicit form can be written as:

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

Therefore, the correct response is:

**an=5(2)n−1**
A sub n is equal to 5 times 2 raised to the n minus 1 power.