Question

Q1: The first three stages of a pattern are shown below. Each stage of the pattern is made up of small squares. Each small square has an area of one square unit. Stage 1 Stage 2 (a) Draw the next two stages of the pattern. (b) The perimeter of Stage 1 of the pattern is four units. The perimeter of Stage 3 Stage 2 of the pattern is 12 units. Find a general formula for the perimeter of Stage n of the pattern, where # EN. (c) Find a general formula for the area of Stage n of the pattern, where E N. (d) What kind of sequence (arithmetic, quadratic, geometric, or none of these) do the areas follow? Justify your answer.​

Answer 1

9514 1404 393

Answer:

a) see attached for the drawings

b) p = 8n -4

c) a(n) = 4n(n -3) +13 for n > 1; a(1) = 1

d) none

Explanation:

a) The overall dimensions of the next two stages of the pattern will be 7×7 and 9×9. Excluding the 4 "ears", the central squares are 5×5 and 7×7. The drawings are shown in the attachment.

__

b) The sequence of perimeter values is ...

4, 12, 20, 28, 36, ...

This is an arithmetic sequence with a common difference of 8. A suitable formula for the perimeter p of stage n is ...

p = 8n -4

__

c) The sequence of area values is ...

1, 5, 13, 29, 53, ...

The first differences are ...

4, 8, 16, 24, ...

and the second differences are ...

4, 8, 8, ...

At no level are the differences "common", so this sequence cannot be described by a polynomial function. After the first one, the second differences are common, so the sequence after the first term can be described by a quadratic function.

The function describing the sequence must be defined in two parts:

`a(n)=\begin{cases}1&\text{ for $n = 1$}\\4n(n-3)+13&\text{ for $n>1$}\end{cases}`

__

d) Because the function must be piecewise defined, the area sequence is not any of the types of sequences listed. After the first term, it is a quadratic sequence.

_____

Additional comment

The leading coefficient of the quadratic formula for the area sequence is half the second difference. When 4n² is subtracted from the sequence of areas, the remaining arithmetic sequence has a common difference of -12. In order to bring the total of these quadratic and arithmetic sequences up to the correct area value, 13 must be added. This formula would give a(1) = a(2) = 5, which is not the correct value for a(1). Hence, that must be defined separately.