Question 2 of 17advanced🔧 ApplyNumerical3 marks

Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases. Bottom row: 4, 13, 8, 7, 11, 3, 10, 14, 25

Correct Answer

Step 1: Identify this as a 9-row pyramid (9 elements in bottom row).

Step 2: Apply binomial coefficients: Top = 1×41 \times 4 + 8×138 \times 13 + 28×828 \times 8 + 56×756 \times 7 + 70×1170 \times 11 + 56×356 \times 3 + 28×1028 \times 10 + 8×148 \times 14 + 1×251 \times 25

Step 3: Calculate: = 4 + 104 + 224 + 392 + 770 + 168 + 280 + 112 + 25

Step 4: Sum = 2079

Exercise: Figure it Out (Section 1) | Q: 1 | (Page Page 6)
For More Understanding

Explanation

The chapter establishes that in a number pyramid, the top number equals the sum of each bottom element multiplied by its corresponding binomial coefficient. For a 9-row pyramid, the coefficients are C(8,0), C(8,1), ..., C(8,8) which are 1, 8, 28, 56, 70, 56, 28, 8, 1.

Solution Steps

  1. Step 1: Identify this as a 9-row pyramid (9 elements in bottom row)

  2. Step 2: Apply binomial coefficients: Top = 1×41 \times 4 + 8×138 \times 13 + 28×828 \times 8 + 56×756 \times 7 + 70×1170 \times 11 + 56×356 \times 3 + 28×1028 \times 10 + 8×148 \times 14 + 1×251 \times 25

  3. Step 3: Calculate: = 4 + 104 + 224 + 392 + 770 + 168 + 280 + 112 + 25

  4. Step 4: Sum = 2079

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