Question

introduction to modeling with functions

brandon enters a contest at the gym. he earns 1 point for his first workout. if he works out the next day, he earns double the points of the previous day. brandon creates the graph to show the number of points that can be earned on the nth consecutive day of working out.

how many points will brandon earn on day 8 if he works out each of the previous 7 days?

• 32
• 64
• 128
• 8

Explanation:

Identify the pattern of points

Using the Function Modeling knowledge point
The problem states that Brandon earns \(1\) point for his first workout. Each consecutive day he works out, he earns double the points of the previous day.
Let \(P(n)\) represent the points earned on day \(n\):

• Day 1: \(P(1) = 1\)
• Day 2: \(P(2) = 2\)
• Day 3: \(P(3) = 4\)
• Day 4: \(P(4) = 8\)
• Day 5: \(P(5) = 16\)
• Day 6: \(P(6) = 32\)
• Day 7: \(P(7) = 64\)

Formulate the general term

Using the Function Modeling knowledge point
The sequence of points forms a geometric sequence with a first term \(a_1 = 1\) and a common ratio \(r = 2\).
The general formula for the points earned on the \(n\)-th consecutive day is:

$$P(n) = 1 \cdot 2^{n-1} = 2^{n-1}$$

Calculate the points for day 8

Using the Function Modeling knowledge point
To find the points earned on day 8, substitute \(n = 8\) into the formula:

$$P(8) = 2^{8-1} = 2^7$$

$$P(8) = 128$$

Answer:

• 32
• 64
• 128 (Correct answer)
• 8