Question

A group of students is arranging squares into layers to create a project. The first layer has 5 squares. The second layer has 10 squares. Which formula represents arithmetic explicit formula to determine the number of squares in each layer?

A.a₁ = 5; aₙ = 5 ⋅ aₙ₋₁, n > 0
B.a₁ = 5; aₙ = 5 ⋅ aₙ₊₁, n > 0
C.a₁ = 5; aₙ = 5 + aₙ₋₁, n > 0
D.a₁ = 5; aₙ = 5 + aₙ₊₁, n > 0

Answer 1

The correct explicit formula for the arithmetic sequence describing the number of squares in each layer is a₁ = 5; aₙ = 5 + (n - 1)×5 for n > 0, reflecting the pattern where each layer has 5 more squares than the previous one.

Step-by-step explanation:

To determine the number of squares in each layer using an arithmetic sequence, we need to find a pattern or a rule that relates each term in the sequence. The first layer has 5 squares and the second layer has 10. This means each layer has 5 more squares than the previous one, indicating the common difference in our arithmetic sequence is 5.

When writing an explicit formula for an arithmetic sequence, the formula typically takes the form aₙ = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference, and n is the term number.

With this pattern, we can write the explicit formula as follows:

• a₁ = 5; this is the number of squares in the first layer.
• aₙ = 5 + (n - 1)×5; this accounts for the 5 additional squares added with each successive layer.
• The correct answer is then: C.a₁ = 5; aₙ = 5 + (n - 1)×5, n > 0, since we are adding 5 times the number of layers minus 1 to the first term to get the number of squares in the nth layer.