Looking at the image, I can see a sequence of four figures labeled 1st, 2nd, 3rd, and 4th. Each figure is made up of diamond shapes (rhombuses) arranged in a pattern.
Let me count the number of diamonds in each figure:
- 1st figure: 1 diamond
- 2nd figure: 4 diamonds
- 3rd figure: 7 diamonds
- 4th figure: 10 diamonds
I can see a clear pattern here. The number of diamonds increases by 3 for each subsequent figure:
- From 1st to 2nd: 1 + 3 = 4
- From 2nd to 3rd: 4 + 3 = 7
- From 3rd to 4th: 7 + 3 = 10
This suggests that the number of diamonds in the nth figure follows an arithmetic sequence with a common difference of 3.
The general formula for an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an is the nth term
- a1 is the first term
- d is the common difference
- n is the term number
In this case:
- a1 = 1 (first figure has 1 diamond)
- d = 3 (common difference)
So the formula for the number of diamonds in the nth figure is:
an = 1 + (n - 1) × 3
an = 1 + 3n - 3
an = 3n - 2
Therefore, the number of diamonds in the nth figure is given by the formula 3n - 2.
To verify:
- For n = 1: 3(1) - 2 = 1 ✓
- For n = 2: 3(2) - 2 = 4 ✓
- For n = 3: 3(3) - 2 = 7 ✓
- For n = 4: 3(4) - 2 = 10 ✓
The solution to the problem is that the number of diamonds in the nth figure is 3n - 2.
Parent Tip: Review the logic above to help your child master the concept of spatial patterns worksheet.
