Question
When a rubber ball is dropped it bounces back up to a height 34
as high as its previous height. You drop this ball from a height of 16 feet and measure its height after each bounce.
Which formula represents a geometric sequence that describes the height of the ball after the number of bounces n
?
(1 point)
Responses
an=16(34)n−1
A sub n is equal to 16 times 3 fourths raised to the n minus 1 power
an=34(16)n−1
A sub n is equal to 3 fourths times 16 raised to the n minus 1 power
an=16+(34)n−1
A sub n is equal to 16 plus 3 fourths raised to the n minus 1 power
an=(n−1)(34)(16)
as high as its previous height. You drop this ball from a height of 16 feet and measure its height after each bounce.
Which formula represents a geometric sequence that describes the height of the ball after the number of bounces n
?
(1 point)
Responses
an=16(34)n−1
A sub n is equal to 16 times 3 fourths raised to the n minus 1 power
an=34(16)n−1
A sub n is equal to 3 fourths times 16 raised to the n minus 1 power
an=16+(34)n−1
A sub n is equal to 16 plus 3 fourths raised to the n minus 1 power
an=(n−1)(34)(16)
Answers
GPT-4o mini
To represent the height of the ball after each bounce as a geometric sequence, we can use the following formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \( a_n \) is the height after \( n \) bounces,
- \( a_1 \) is the initial height,
- \( r \) is the common ratio (the factor by which the height changes with each bounce),
- \( n \) is the number of bounces.
In this case:
- The initial height \( a_1 = 16 \) feet (the height from which the ball is dropped),
- The common ratio \( r = \frac{3}{4} \) (the ball bounces back to \( \frac{3}{4} \) of its previous height).
Plugging these values into the formula gives:
\[
a_n = 16 \left(\frac{3}{4}\right)^{n-1}
\]
This matches the first option in your responses:
**an = 16(3/4)^(n - 1)**
So, the correct answer is:
**A sub n is equal to 16 times 3 fourths raised to the n minus 1 power.**
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \( a_n \) is the height after \( n \) bounces,
- \( a_1 \) is the initial height,
- \( r \) is the common ratio (the factor by which the height changes with each bounce),
- \( n \) is the number of bounces.
In this case:
- The initial height \( a_1 = 16 \) feet (the height from which the ball is dropped),
- The common ratio \( r = \frac{3}{4} \) (the ball bounces back to \( \frac{3}{4} \) of its previous height).
Plugging these values into the formula gives:
\[
a_n = 16 \left(\frac{3}{4}\right)^{n-1}
\]
This matches the first option in your responses:
**an = 16(3/4)^(n - 1)**
So, the correct answer is:
**A sub n is equal to 16 times 3 fourths raised to the n minus 1 power.**