Question

what is the shading style of the twenty - third figure in the sequence below? the twenty - third figure in the sequence is filled. striped. unshaded.

Explanation:

Answer:

First, identify the pattern in the sequence. Let's list the figures with their positions:

1: Unshaded (Circle)
2: Filled (Triangle)
3: Striped (Square)
4: Unshaded (Star)
5: Filled (Circle)
6: Striped (Triangle)
7: Unshaded (Square)
8: Filled (Star)

Wait, actually, let's check the pattern of shading and shape. Wait, maybe the pattern repeats every 4? Wait, no, let's check the shading cycle. Wait, positions 1,4,7: unshaded (1: circle, 4: star, 7: square) – every 3? No, 1,4,7: difference 3. 2,5,8: filled (2: triangle, 5: circle, 8: star) – difference 3. 3,6: striped (3: square, 6: triangle) – wait, 3,6,9? Wait, 3: striped, 6: striped, 9:? Let's count the positions:

Position: 1 2 3 4 5 6 7 8
Shading: U F S U F S U F

Ah! Wait, 1: Unshaded (U), 2: Filled (F), 3: Striped (S), 4: Unshaded (U), 5: Filled (F), 6: Striped (S), 7: Unshaded (U), 8: Filled (F)... So the cycle is 3: U, F, S, repeating every 3? Wait, 1: U, 2:F, 3:S, 4:U (1+3), 5:F (2+3), 6:S (3+3), 7:U (4+3=7), 8:F (5+3=8), 9:S (6+3=9), etc. So the cycle length is 3: [U, F, S], [U, F, S], [U, F, S],...

To find the 23rd term, divide 23 by 3. 23 ÷ 3 = 7 with a remainder of 2 (since 3×7=21, 23-21=2). So the remainder is 2. In the cycle [U, F, S], the 2nd term is F (filled). Wait, but let's check:

Term 1: remainder 1 → U
Term 2: remainder 2 → F
Term 3: remainder 0 → S (since 3÷3=1 remainder 0)
Term 4: 4÷3=1 remainder 1 → U
Term 5: 5÷3=1 remainder 2 → F
Term 6: 6÷3=2 remainder 0 → S
Yes, so remainder 1: U, remainder 2: F, remainder 0: S.

23 divided by 3: 3×7=21, 23-21=2. So remainder 2. Therefore, the 23rd figure has shading style "filled".

Wait, but let's confirm with the sequence:

1: U (circle)
2: F (triangle)
3: S (square)
4: U (star)
5: F (circle)
6: S (triangle)
7: U (square)
8: F (star)
9: S (circle)
10: U (triangle)
11: F (square)
12: S (star)
13: U (circle)
14: F (triangle)
15: S (square)
16: U (star)
17: F (circle)
18: S (triangle)
19: U (square)
20: F (star)
21: S (circle)
22: U (triangle)
23: F (square)? Wait, no, maybe the shape pattern is different. Wait, the shapes: circle, triangle, square, star, circle, triangle, square, star... So shape cycle is 4: circle (1), triangle (2), square (3), star (4), circle (5), triangle (6), square (7), star (8), circle (9), triangle (10), square (11), star (12), circle (13), triangle (14), square (15), star (16), circle (17), triangle (18), square (19), star (20), circle (21), triangle (22), square (23). Wait, shape at 23: square (since 23 ÷4=5 remainder 3; 4×5=20, 23-20=3; 3rd shape is square). Now shading: let's check shading per shape? Wait, no, earlier I thought shading cycle is 3, but maybe shading is per position. Wait, position 1: U (circle), 2:F (triangle), 3:S (square), 4:U (star), 5:F (circle), 6:S (triangle), 7:U (square), 8:F (star), 9:S (circle), 10:U (triangle), 11:F (square), 12:S (star), 13:U (circle), 14:F (triangle), 15:S (square), 16:U (star), 17:F (circle), 18:S (triangle), 19:U (square), 20:F (star), 21:S (circle), 22:U (triangle), 23:F (square). So shading at 23: Filled.

Yes, so the 23rd figure is filled.

filled.