Algebra Play · Lesson 4 of 10
Patterns in Number Pyramids
“Generalise number pyramids with algebraic top expressions and explore their connection with the Virahāṅka–Fibonacci sequence.”
• Derive expressions for the top of three-row and four-row number pyramids. • Explain why repeated addition creates coefficient patterns such as 1, 2, 1 and 1, 3, 3, 1. • Use a top-row expression to find answers without building every intermediate row. • Investigate what happens when consecutive Virahāṅka–Fibonacci numbers form the bottom row. • Make and justify a general prediction for an n-row pyramid built from this sequence.
Once the pyramid rule is understood, we can stop treating every pyramid as a separate puzzle. Algebra lets us describe the whole structure in one expression. This is especially useful when a pyramid has many rows, because the same lower value may influence the top several times.
From a Diagram to a General Expression
For a two-row pyramid with bottom values a and b, the top is simply a + b. For a three-row pyramid with bottom values a, b, c, the second row is a + b and b + c, so the top becomes (a + b) + (b + c) = a + 2b + c.
| Rows | Bottom row | Top expression |
|---|---|---|
| 2 | a, b | a + b |
| 3 | a, b, c | a + 2b + c |
| 4 | a, b, c, d | a + 3b + 3c + d |
Problem
Show why the top of a four-row pyramid with bottom a, b, c, d is a + 3b + 3c + d.
- 1.The row above the bottom is a+b, b+c, c+d.
- 2.The next row is (a+b)+(b+c)=a+2b+c and (b+c)+(c+d)=b+2c+d.
- 3.Add those two expressions for the top.
- 4.(a+2b+c) + (b+2c+d) = a+3b+3c+d.
- 5.The inner bottom values influence more paths to the top, so they receive larger coefficients.
Why the Coefficients Form a Pattern
The coefficients are not arbitrary. Each coefficient counts how many addition paths carry a bottom value to the top. In a three-row pyramid, the middle value b reaches the top through two paths. In a four-row pyramid, b and c each reach the top through three paths.
The coefficient rows begin 1,1; then 1,2,1; then 1,3,3,1. Each inner number is the sum of the two numbers just above it in the previous coefficient row. We do not need to name a more advanced triangle of coefficients here; what matters is understanding that repeated addition creates this pattern.
Problem
Find the top of a four-row pyramid with bottom row 4, 13, 8, 7.
- 1.Use top = a + 3b + 3c + d.
- 2.Substitute a=4, b=13, c=8, d=7.
- 3.Top = 4 + 39 + 24 + 7 = 74.
- 4.This gives the same answer as filling all intermediate boxes, but with less work.
Problem
A four-row pyramid has bottom row 5, x, 4, 6 and top 47. Find x.
- 1.Use 5 + 3x + 3(4) + 6 = 47.
- 2.Combine known terms: 23 + 3x = 47.
- 3.So 3x = 24 and x = 8.
- 4.Checking by building the pyramid confirms the top is 47.
Virahāṅka–Fibonacci Numbers in a Pyramid
Consider the sequence 1, 2, 3, 5, 8, 13, ... in which each term equals the sum of the previous two. If consecutive terms are placed along the bottom of a pyramid, the pyramid rule produces later terms of the same sequence.
| Bottom row | Next row | Top |
|---|---|---|
| 1, 2, 3 | 3, 5 | 8 |
| 1, 2, 3, 5 | 3, 5, 8 | 21 after continuing upward |
Why does this happen? If two adjacent entries are consecutive sequence terms, their sum is the next term. So every row above is again made from consecutive terms, but the starting position in the sequence jumps forward.
Problem
Place 1, 2, 3, 5 in a four-row pyramid. What appears?
- 1.First row above: 1+2=3, 2+3=5, 3+5=8, giving 3, 5, 8.
- 2.Next row: 3+5=8 and 5+8=13, giving 8, 13.
- 3.Top: 8+13=21.
- 4.Every entry is another term of the same sequence.
For n = 3, the top is the 5th term, 8. For n = 4, it is the 7th term, 21. The rule follows because moving up one row replaces adjacent terms by their sum, which advances the sequence index by two.
Try n = 5 using bottom row 1, 2, 3, 5, 8. Predict the top before completing the pyramid, then verify it.
Using Patterns Responsibly
A pattern seen in a few examples is a conjecture. To trust it for a very large pyramid, we need a reason. Here the reason comes from the defining sequence rule: every pair of consecutive terms sums to the next term. That property is preserved row after row.
Do not assume that every list of increasing numbers behaves like the Virahāṅka–Fibonacci sequence. The pyramid remains within the sequence only because each adjacent pair already satisfies the sequence's addition rule.
Quiz
What is the top expression for a four-row pyramid with bottom a,b,c,d?
Why does b have coefficient 3 in the four-row top expression?
With bottom row 1,2,3, what is the top?
If the first four sequence terms 1,2,3,5 are used, what is the top?
For an n-row pyramid using the first n Virahāṅka–Fibonacci terms, which sequence term is predicted at the top?
Practice Problems
- Find the top of a four-row pyramid with bottom row 8, 19, 21, 13 without filling every box.
- Write and simplify the top expression for bottom row p, q, r, s.
- A four-row pyramid has bottom 7, x, 5, 2 and top 48. Find x.
- Build the pyramid with bottom 1,2,3,5,8 and identify the top.
- Explain why each row in a sequence pyramid still contains consecutive Virahāṅka–Fibonacci terms.
- Use the formula for the top index to state which sequence term appears at the top of a 29-row pyramid.
Use 8 + 3(19) + 3(21) + 13 = 8 + 57 + 63 + 13 = 141.
Key Takeaways
• Repeated pyramid addition can be compressed into one algebraic top expression. • A four-row pyramid has coefficients 1, 3, 3, 1 across the bottom. • Coefficients count how many addition paths carry each bottom value to the top. • Consecutive Virahāṅka–Fibonacci terms remain within the same sequence as the pyramid is built upward. • For an n-row pyramid using the first n sequence terms, the top is the (2n−1)th term.