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 $a_n=2a_{n-1}$ into an explicit form, we can calculate the first few terms:

• $a_1=5$
• $a_2=2a_1=2\cdot 5=10$
• $a_3=2a_2=2\cdot 10=20$
• $a_4=2a_3=2\cdot 20=40$

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:

$$a_n=5(2{)}^{n-1}$$