Question

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:

for each statement:

Statement 1: Height with 5 bowls

The first bowl is 2.5 in. Each additional bowl (4 additional bowls for 5 total) adds 1 in. So height = 2.5 + (5 - 1)*1 = 2.5 + 4 = 6.5 in. So this is true.

Statement 2: Explicit rule

For an arithmetic sequence, explicit rule is \(a_n=a_1+(n - 1)d\), where \(a_1 = 2.5\), \(d = 1\). So \(a_n=2.5+(n - 1)\times1=2.5+(n - 1)\). True.

Statement 3: Recursive rule

Recursive rule for 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 \(a_1 = 2.5\)). True.

Statement 4: Domain and range

The number of bowls \(n\) is a positive integer (1, 2, 3,...), so domain is discrete. The height is calculated for integer number of bowls, so range is also discrete (2.5, 3.5, 4.5,...). True.

Statement 5: Linear equation \(h(b)=2.5 + b\)

If \(b\) is number of bowls, the first bowl (\(b = 1\)) should be 2.5, but \(h(1)=2.5 + 1=3.5\), which is wrong. The correct equation is \(h(b)=2.5+(b - 1)=1.5 + b\). So this is false.

Answer:

• 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.