Question: 5) Study the pattern carefully.
Pattern 1
Pattern 2
Pattern 3
Pattern 4
Pattern
Number
Number of
Shaded Square
Number of
Unshaded Sq
Total no.
Of squares
1
1
3
4
2
6
3
9
3
6
10
16
4
15
10
25
:
:
:
:
8
45
36
81
a) Find the total number of squares needed to make Pattern 16.
b) If a total of 400 squares are used to form a new pattern, which pattern is it?
c) Find the total number of shaded squares needed to make Pattern 16?
Pattern 1 is a 2\times2 square with 1 shaded square in the bottom-right position and 3 unshaded squares, position_hint = top-center.
Pattern 2 is a 3\times3 arrangement with a shaded top row and left column forming an L-shape, plus a shaded bottom-right square; the center and two right-middle squares are unshaded, position_hint = top-center.
Pattern 3 is a 4\times4 arrangement with a shaded top row and left column, a shaded vertical bar in the second column, and a shaded middle-right block, position_hint = top-center.
Pattern 4 is a 5\times5 arrangement with a shaded top row and left column, continuing the inward L-shaped shaded path with a shaded central vertical bar and bottom-right shaded square, position_hint = top-center.
1. **State the problem:** We have a sequence of square patterns with given numbers of shaded and unshaded squares and total squares. We want to find:
a) Total squares in Pattern 16.
b) Which pattern corresponds to 400 total squares.
c) Number of shaded squares in Pattern 16.
2. **Analyze the total number of squares:** From the table, total squares for Pattern $n$ are:
Pattern 1: 4 = $2^2$
Pattern 2: 9 = $3^2$
Pattern 3: 16 = $4^2$
Pattern 4: 25 = $5^2$
Pattern 8: 81 = $9^2$
We see total squares = $(n+1)^2$.
3. **Find total squares for Pattern 16:**
$$\text{Total squares} = (16+1)^2 = 17^2 = 289$$
4. **Find pattern number for total squares = 400:**
We want $ (n+1)^2 = 400$.
Taking square root:
$$n+1 = \sqrt{400} = 20$$
$$n = 20 - 1 = 19$$
So, Pattern 19 has 400 total squares.
5. **Find number of shaded squares:** From the table:
Pattern 1 shaded = 1
Pattern 2 shaded = 6
Pattern 3 shaded = 6
Pattern 4 shaded = 15
Pattern 8 shaded = 45
Notice shaded squares correspond to $n^2 - 1$ for $n \geq 2$? Check:
Pattern 2: $2^2 - 1 = 4 - 1 = 3$ (doesn't match 6)
Try another approach: shaded squares for Pattern $n$ seem to be $\frac{n(n+1)}{2}$ (triangular number) times 3?
Check Pattern 4 shaded = 15:
Triangular number $T_5 = \frac{5\times6}{2} = 15$ matches shaded squares.
So shaded squares = triangular number $T_{n+1} = \frac{(n+1)(n+2)}{2}$?
Check Pattern 2:
$T_3 = \frac{3\times4}{2} = 6$ matches shaded squares.
Check Pattern 3:
$T_4 = \frac{4\times5}{2} = 10$ but shaded squares given is 6, so no.
Another pattern: shaded squares for Pattern $n$ = $\frac{n(n+1)}{2}$?
Pattern 2: $\frac{2\times3}{2} = 3$ no.
Pattern 8 shaded = 45, which is $\frac{9\times10}{2} = 45$.
So shaded squares = $\frac{(n+1)(n+2)}{2}$ for Pattern $n$?
Pattern 1 shaded = 1, $\frac{2\times3}{2} = 3$ no.
Try shaded squares = $\frac{n(n+1)}{2}$:
Pattern 1: $\frac{1\times2}{2} = 1$ matches
Pattern 2: $\frac{2\times3}{2} = 3$ given is 6 no
Given data inconsistent for shaded squares, but from Pattern 8 shaded = 45 and total squares = 81 = $9^2$, shaded squares = $\frac{9\times10}{2} = 45$ matches.
So shaded squares for Pattern $n$ = $\frac{(n+1)(n+2)}{2}$.
6. **Calculate shaded squares for Pattern 16:**
$$\text{Shaded squares} = \frac{(16+1)(16+2)}{2} = \frac{17 \times 18}{2} = 17 \times 9 = 153$$
**Final answers:**
- a) Total squares in Pattern 16: $289$
- b) Pattern with 400 squares: Pattern $19$
- c) Shaded squares in Pattern 16: $153$