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NCERT Solutions Ganita Prakash (Part 2) Chapter 6 .3 Number Pyramids — Figure it Out

Book page 1406 Updated on2026-09-05

Q1.
Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases. (i) 4, 13, 8 (ii) 7, 11, 3 (iii) 10, 14, 25
Answer

Use the three-row formula top = a + 2b + c, doubling only the middle entry.

(i) 4 + 2(13) + 8 = 4 + 26 + 8 = 38
(ii) 7 + 2(11) + 3 = 7 + 22 + 3 = 32
(iii) 10 + 2(14) + 25 = 10 + 28 + 25 = 63
Bottom rowMiddle row (check)Top
4, 13, 817, 2138
7, 11, 318, 1432
10, 14, 2524, 3963
Why it happens: the formula is not a shortcut that skips the pyramid — it is the pyramid, written out once and for all. Because a + 2b + c was derived from letter-numbers, it holds for every bottom row, so there is nothing left to build.
Q2.
Write an expression for the topmost row of a pyramid with 4 rows in terms of the values in the bottom row.
Answer

With bottom row a, b, c, d:

Third row: a + b,   b + c,   c + d
Second row: (a + b) + (b + c) = a + 2b + c   and   (b + c) + (c + d) = b + 2c + d
Top: (a + 2b + c) + (b + 2c + d) = a + 3b + 3c + d
Why it happens: a and d sit at the ends, with just one path each to the top, so they are counted once. From b there are three ways up, and the same from c — which is why both carry a multiplier of 3. Notice also that 1 + 3 + 3 + 1 = 8 = 2³: each extra row doubles the total pull of the bottom row, because every box is copied into the two boxes above it.
Tip: the pattern continues. Five rows give a + 4b + 6c + 4d + e, and each new list of multipliers comes from adding neighbours in the previous list.
Q3.
Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases. (i) 8, 19, 21, 13 (ii) 7, 18, 19, 6 (iii) 9, 7, 5, 11
Answer

Use top = a + 3b + 3c + d.

(i) 8 + 3(19) + 3(21) + 13 = 8 + 57 + 63 + 13 = 141
(ii) 7 + 3(18) + 3(19) + 6 = 7 + 54 + 57 + 6 = 124
(iii) 9 + 3(7) + 3(5) + 11 = 9 + 21 + 15 + 11 = 56
Bottom rowThird rowSecond rowTop
8, 19, 21, 1327, 40, 3467, 74141
7, 18, 19, 625, 37, 2562, 62124
9, 7, 5, 1116, 12, 1628, 2856
Check it yourself: in (ii) and (iii) the second row came out with two equal boxes. A quick shortcut for those: the top is just twice that number — 2 × 62 = 124 and 2 × 28 = 56 ✓.
Q4.
If the first three Virahāṅka-Fibonacci numbers are written in the bottom row of a number pyramid with three rows, fill in the rest of the pyramid. What numbers appear in the grid? What is the number at the top? Are they all Virahāṅka-Fibonacci numbers?
Answer

The first three Virahāṅka-Fibonacci numbers are 1, 2, 3.

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Bottom 1, 2, 3 → middle 3, 5 → top 8.
Middle row: 1 + 2 = 3,   2 + 3 = 5
Top: 3 + 5 = 8

The numbers appearing are 1, 2, 3, 3, 5, 8. The number at the top is 8. Yes — every one of them is a Virahāṅka-Fibonacci number, and 8 is the 5th one in the sequence 1, 2, 3, 5, 8.

Why it happens: the pyramid rule and the Virahāṅka-Fibonacci rule are the same rule. Adding two neighbours in 1, 2, 3, 5, 8, … gives the next term of the sequence. So the middle row 1 + 2, 2 + 3 is just 3, 5 — the sequence again, but starting two places later. Adding once more gives 8.
Q5.
What can you say about the numbers in the pyramid and the number at the top in the following cases? (i) The first four Virahāṅka-Fibonacci numbers are written in the bottom row of a four row pyramid. (ii) The first 29 Virahāṅka-Fibonacci numbers are written in the bottom row of a 29 row pyramid.
Answer

(i) The bottom row is 1, 2, 3, 5.

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Bottom 1, 2, 3, 5 → 3, 5, 8 → 8, 13 → top 21.

Every row is again a run of consecutive Virahāṅka-Fibonacci numbers, each row starting two places later than the row below. The top is 21, the 7th Virahāṅka-Fibonacci number.

Row (from the bottom)NumbersPosition in the sequence
1 (bottom)1, 2, 3, 51st to 4th
23, 5, 83rd to 5th
38, 135th to 6th
4 (top)217th

(ii) The same thing happens, only for longer. Every number in a 29-row pyramid is a Virahāṅka-Fibonacci number; row r counted from the bottom holds consecutive terms starting at the (2r – 1)th. The top is the (2 × 29 – 1) = 57th Virahāṅka-Fibonacci number, which is

Top = 591 286 729 879
Why it happens: write the bottom row as F₁, F₂, …, F₂₉. Adding neighbours gives F₁ + F₂ = F₃, F₂ + F₃ = F₄, and so on, so the second row is F₃, F₄, …, F₃₀. Each row up loses one entry and starts two places further along. After 28 steps only one entry is left, and it starts at position 1 + 2 × 28 = 57.
Tip: you never have to build 29 rows to answer this. Following the position in the sequence instead of the numbers themselves is what makes the question easy.
Q6.
If the bottom row of an n row pyramid contains the first n Virahāṅka-Fibonacci numbers, what can we say about the numbers in the pyramid? What can we say about the number at the top?
Answer

Every number in the pyramid is a Virahāṅka-Fibonacci number, and the top is the (2n – 1)th one.

Bottom row (row 1): F₁, F₂, …, Fn
Row 2: F₃, F₄, …, Fn+1
Row 3: F₅, F₆, …, Fn+2
Row r: F2r–1, F2r, …, Fn+r–1  (that is n – r + 1 numbers)
Row n (the top): the single number F2n–1
Why it happens: the defining rule Fj + Fj+1 = Fj+2 is exactly the pyramid rule. So adding neighbours along a run of consecutive terms produces another run of consecutive terms — one entry shorter, and starting two places later, because the first sum Fj + Fj+1 lands on Fj+2. Going up n – 1 rows moves the starting position from 1 to 1 + 2(n – 1) = 2n – 1.
nBottom rowTopPosition 2n – 1
21, 233rd
31, 2, 385th
41, 2, 3, 5217th
51, 2, 3, 5, 8559th
Check it yourself: for n = 5 the formula a + 4b + 6c + 4d + e gives 1 + 8 + 18 + 20 + 8 = 55, and 55 is indeed the 9th number in 1, 2, 3, 5, 8, 13, 21, 34, 55. ✓
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