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 minus 1 power

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

Answer 1

To convert the recursive sequence given by \( a_1 = 5 \) and \( a_n = 2a_{n-1} \) into explicit form, we can start by calculating a few terms to identify a pattern.

1. \( a_1 = 5 \)
2. \( a_2 = 2a_1 = 2 \times 5 = 10 \)
3. \( a_3 = 2a_2 = 2 \times 10 = 20 \)
4. \( a_4 = 2a_3 = 2 \times 20 = 40 \)
5. \( a_5 = 2a_4 = 2 \times 40 = 80 \)

Now let's observe the terms:

• \( a_1 = 5 = 5 \times 2^0 \)
• \( a_2 = 10 = 5 \times 2^1 \)
• \( a_3 = 20 = 5 \times 2^2 \)
• \( a_4 = 40 = 5 \times 2^3 \)
• \( a_5 = 80 = 5 \times 2^4 \)

From the pattern, we see that:

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

Thus, the explicit form of the recursive sequence is:

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

This corresponds to the first response: A sub n is equal to 5 times 2 raised to the n minus 1 power.