Convert the recursive form a1=5, an=2an−1

Question

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=3(7)n
A sub n is equal to 3 times 7 to the n th power

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

Answer 1

To convert the recursive form given by \( a_1 = 5 \) and \( a_n = 2a_{n-1} \) to an explicit form, we can first observe the pattern in the terms:

1. \( a_1 = 5 \)
2. \( a_2 = 2a_1 = 2 \times 5 = 10 \)
3. \( a_3 = 2a_2 = 2 \times 10 = 20 \)
4. \( a_4 = 2a_3 = 2 \times 20 = 40 \)

From these calculations, we can see that:

• \( a_1 = 5 \)
• \( a_2 = 5 \times 2^1 = 10 \)
• \( a_3 = 5 \times 2^2 = 20 \)
• \( a_4 = 5 \times 2^3 = 40 \)

Following this pattern, we can deduce the explicit formula:

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

Now we can match this with the given options:

• (4) \( a_n = 5(2)^{n-1} \)
  • This matches with our derived explicit form.

Thus, the correct response is:

\( a_n = 5(2)^{n-1} \)
A sub n is equal to 5 times 2 raised to the n minus 1 power.