Question

check your understanding
the first bowl of a stack of nested bowls is 2.5 inches (in.) tall. each bowl that is added to the stack adds 1 inch to the height of the stack.
select all the true statements.
□ the height of the stack with 5 bowls would be 6.5 inches.
□ this situation can be modeled by the explicit rule $a_n = 2.5 + (n - 1)$.
□ this situation could be modeled by the recursive rule $a_n = 1 + a_{n - 1}$.
□ the domain and range of the function that represents this sequence is discrete
□ this situation could be modeled by the linear equation $h(b) = 2.5 + b$, where $h$ is the height of the stack and $b$ is the number of bowls in the stack.

Explanation:

Step1: Analyze the first statement

The first bowl is 2.5 inches, and each additional bowl adds 1 inch. For 5 bowls, the number of additional bowls is \(5 - 1 = 4\). So the height is \(2.5+4\times1 = 6.5\) inches. So this statement is true.

Step2: Analyze the second statement

The explicit formula for an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_1 = 2.5\) and \(d = 1\). So \(a_n=2.5+(n - 1)\times1=2.5+(n - 1)\). This statement is true.

Step3: Analyze the third statement

The recursive formula for an arithmetic sequence is \(a_n=a_{n - 1}+d\), here \(d = 1\) and \(a_1 = 2.5\). So \(a_n=1 + a_{n - 1}\) (with \(a_1 = 2.5\)). This statement is true.

Step4: Analyze the fourth statement

The number of bowls \(n\) is a positive integer (discrete), and the height \(a_n\) is also a discrete value (since we add 1 inch for each bowl, the height will be in increments of 1 from 2.5). So the domain (number of bowls) and range (height) are discrete. This statement is true.

Step5: Analyze the fifth statement

If \(b\) is the number of bowls, the height \(h(b)=2.5+(b - 1)=1.5 + b\), not \(2.5 + b\). So this statement is false.

Answer:

The true statements are:

• The height of the stack with 5 bowls would be 6.5 inches.
• This situation can be modeled by the explicit rule \(a_n = 2.5+(n - 1)\).
• This situation could be modeled by the recursive rule \(a_n = 1+a_{n - 1}\).
• The domain and range of the function that represents this sequence is discrete.