Question

A fashion designer is designing a fabric pattern. In each row, there are some shaded squares and unshaded triangles.

Based on the given figure, answer the following questions.

1. Identify the AP for the number of squares in each row.
2. Identify the AP for the number of triangles in each row.
3. If each shaded square is of side 2 cm then find the shaded area when 15 rows have been designed.
4. Write a formula for finding the total number of triangles in n number of rows. Hence, find \(S_{10}\).

Answer 1

From the given fabric pattern, we observe each row from top to bottom:

Row 1: 1 square, 2 triangles

Row 2: 3 squares, 6 triangles

Row 3: 5 squares, 10 triangles

Row 4: 7 squares, 14 triangles

i. AP for the number of squares in each row:

The sequence representing squares in each row is: \[1, 3, 5, 7, \ldots\] 

Here, first term \[a = 1\], common difference \[d = 3 - 1 = 2\].

\[1, 3, 5, 7, \ldots\]

ii. AP for the number of triangles in each row:

The sequence representing triangles in each row is: \[2, 6, 10, 14, \ldots\] 

Here, first term [a = 2], common difference \[d = 6 - 2 = 4\].

\[2, 6, 10, 14, \ldots\]

iii. Shaded area when 15 rows have been designed:

Total number of squares in 15 rows: \[S_n = \frac{n}{2}[2a + (n - 1)d]\] 

For squares: \[a = 1, d = 2, n = 15\] 

\[S_{15} = \frac{15}{2}[2(1) + (15 - 1)2]\] 

\[= \frac{15}{2}[2 + 28]\] 

\[= \frac{15}{2} \times 30\] 

\[= 225\] 

Alternatively, sum of first n odd positive integers is \[n^2 = 15^2 = 225\].

Area of one square: \[\text{Area} = \text{side}^2 = 2\text{ cm} \times 2\text{ cm} = 4\text{ cm}^2\]

Total shaded area: \[\text{Total Area} = 225 \times 4\text{ cm}^2 = 900\text{ cm}^2\]

The total shaded area is \[900\text{ cm}^2\].

iv. Formula for total triangles in n rows and \(S_{10}\):

For triangles: \[a = 2, d = 4\] 

\[S_n = \frac{n}{2}[2(2) + (n - 1)4]\] 

\[S_n = \frac{n}{2}[4 + 4n - 4]\]

\[= \frac{n}{2}[4n]\]

\[= 2n^2\]

Finding \(S_{10}\): \[S_{10} = 2(10)^2\]

\[= 2 \times 100\]

\[= 200\] 

Note: If taking single-row triangular units as \[1, 3, 5, \ldots\], sum is \[n^2 = 100\]; for the AP \[2, 6, 10, \ldots\], 

\[S_{10} = 200\]

Formula is \[S_n = 2n^2\] or \[n^2\] for pairs, giving \[S_{10} = 100\].