Let’s solve this step by step.
We are given Pascal’s Triangle, and we need to fill it in using the rule:
- The leftmost and rightmost numbers in each row are always 1.
- Every other number is the sum of the two numbers directly above it (one from the upper left, one from the upper right).
The triangle starts like this:
Row 1: 1
Row 2: 1 1
Row 3: 1 2 1
Row 4: 1 3 3 1
Row 5: 1 4 6 4 1
Row 6: 1 5 10 10 5 1
Row 7: 1 6 15 20 15 6 1
Row 8: 1 7 21 35 35 21 7 1
Row 9: 1 8 28 56 70 56 28 8 1
Row 10: 1 9 36 84 126 126 84 36 9 1
Row 11:1 10 45 120 210 252 210 120 45 10 1
Row 12:1 11 55 165 330 462 462 330 165 55 11 1
Now, let’s compute the
sum of each row:
Row 1: 1 → sum =
1
Row 2: 1 + 1 =
2
Row 3: 1 + 2 + 1 =
4
Row 4: 1 + 3 + 3 + 1 =
8
Row 5: 1 + 4 + 6 + 4 + 1 =
16
Row 6: 1 + 5 + 10 + 10 + 5 + 1 =
32
Row 7: 1 + 6 + 15 + 20 + 15 + 6 + 1 =
64
Row 8: 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 =
128
Row 9: 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 =
256
Row 10: 1 + 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1 =
512
Row 11: 1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 =
1024
Row 12: 1 + 11 + 55 + 165 + 330 + 462 + 462 + 330 + 165 + 55 + 11 + 1 =
2048
You can see a pattern: each row sum is double the previous one. That’s because the sums follow powers of 2:
Row n sum = 2^(n-1)
Check:
Row 1: 2^0 = 1
✔
Row 2: 2^1 = 2
✔
Row 3: 2^2 = 4
✔
...
Row 12: 2^11 = 2048
✔
So now we fill in the blanks:
The sum of row 1:
1
The sum of row 2:
2
The sum of row 3:
4
The sum of row 4:
8
The sum of row 5:
16
The sum of row 6:
32
The sum of row 7:
64
The sum of row 8:
128
The sum of row 9:
256
The sum of row 10:
512
The sum of row 11:
1024
The sum of row 12:
2048
What do you notice? Each row sum is twice the previous row’s sum — or equivalently, they are powers of 2 starting from 2⁰.
Final Answer:
The sum of row 1: 1
The sum of row 2: 2
The sum of row 3: 4
The sum of row 4: 8
The sum of row 5: 16
The sum of row 6: 32
The sum of row 7: 64
The sum of row 8: 128
The sum of row 9: 256
The sum of row 10: 512
The sum of row 11: 1024
The sum of row 12: 2048
Parent Tip: Review the logic above to help your child master the concept of pascal triangle worksheet with answers.
