Question

8.. the pattern below shows the number of triangles growing in each rug row (geometric):
row 1: ▲
row 2: ▲▲
row 3: ▲▲▲▲
row 4: ▲▲▲▲▲▲▲▲
find the formula and total number of triangles after 8 rows.

9.a master weaver is designing a large, rectangular two grey hills style rug. the central design consists of a sequence of stacked diamond shapes. the weaver tracks the number of black threads needed for the vertical column of the diamond at the center of each row. the number of black threads per row forms the following sequence: a₁, a₂, a₃,……aₙ. after analyzing the first five rows, the weaver notes the sequence is 15, 18, 21, 24, 27, ...
question: given the arithmetic sequence representing the black threads per row, analyze the properties of the series, synthesize this analysis with the general form of the arithmetic series sum ($s_n = \frac{n}{2}(2a_1 + (n - 1)d)$), and then derive an equivalent, simplified recursive formula for the total number of black threads required for any rug with n such pattern rows. justify how this recursive formula effectively models the accumulation of complexity (total threads) as the rug size (n) increases, maintaining the established patterns integrity.

Explanation:

Problem 8

Step1: Identify the pattern

Row 1: \(1 = 2^0\) triangle
Row 2: \(2 = 2^1\) triangles
Row 3: \(4 = 2^2\) triangles
Row 4: \(8 = 2^3\) triangles
So, the number of triangles in row \(n\) is \(a_n = 2^{n - 1}\).

Step2: Find the total number of triangles after \(n\) rows

This is a geometric series with first term \(a_1 = 1\) and common ratio \(r = 2\). The sum of a geometric series is given by \(S_n=\frac{a_1(r^n - 1)}{r - 1}\) (since \(r
eq1\)).

For \(n = 8\):
\(S_8=\frac{1\times(2^8 - 1)}{2 - 1}\)
\(= 2^8 - 1\)
\(= 256 - 1 = 255\)

Step1: Identify the arithmetic sequence properties

Given the sequence \(15, 18, 21, 24, 27, \dots\)
First term, \(a_1 = 15\)
Common difference, \(d = 18 - 15 = 3\)

Step2: Recall the arithmetic series sum formula

The sum of the first \(n\) terms of an arithmetic series is \(S_n=\frac{n}{2}(2a_1+(n - 1)d)\).

Substitute \(a_1 = 15\) and \(d = 3\):
\(S_n=\frac{n}{2}(2\times15+(n - 1)\times3)\)
\(=\frac{n}{2}(30 + 3n - 3)\)
\(=\frac{n}{2}(3n + 27)\)
\(=\frac{n(3n + 27)}{2}\)
\(=\frac{3n(n + 9)}{2}\) or simplify further: \(S_n=\frac{3}{2}n^2+\frac{27}{2}n\)

Step3: Derive the recursive formula

For an arithmetic series, the recursive formula for the sum \(S_n\) can be derived from \(S_n = S_{n - 1}+a_n\).
First, find \(a_n\): \(a_n=a_1+(n - 1)d = 15+(n - 1)\times3 = 3n + 12\)
So, \(S_n = S_{n - 1}+3n + 12\) with \(S_1 = 15\) (base case).

Step4: Justify the recursive formula

As \(n\) increases, each new row adds \(3\) more threads than the previous row (since \(d = 3\)). The recursive formula \(S_n = S_{n - 1}+a_n\) models this by adding the number of threads in the \(n\)-th row (\(a_n\)) to the total threads of the first \(n - 1\) rows (\(S_{n - 1}\)). This ensures the pattern’s integrity because each row’s thread count follows the arithmetic progression, and the total accumulates by adding the current row’s count to the prior total.

Answer:

Formula for the number of triangles in row \(n\): \(a_n = 2^{n - 1}\)
Total number of triangles after 8 rows: \(\boldsymbol{255}\)

Problem 9