NCERT Solutions Ganita Prakash Chapter 1 Figure it Out — Visualising Number Sequences
Book page 5 Updated on2026-09-05
Q1.
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
Answer
The sixth picture of each sequence is described below — draw it just after the fifth picture of Table 2.
Sequence
Picture already drawn
Next picture to draw
All 1's
1, 1, 1, 1, 1
one more single dot → 1
Counting numbers
1, 2, 3, 4, 5
a line of 6 dots → 6
Odd numbers
1, 3, 5, 7, 9
an “L / T” shape of 11 dots → 11
Even numbers
2, 4, 6, 8, 10
two rows of 6 dots → 12
Triangular numbers
1, 3, 6, 10, 15
a triangle of 6 rows → 21
Squares
1, 4, 9, 16, 25
a 6 × 6 square of dots → 36
Cubes
1, 8, 27, 64, 125
a 6 × 6 × 6 cube of dots → 216
Triangular numbers 1, 3, 6, 10, 15 — each new picture adds one more row at the bottom. The next picture has a 6-dot row, giving 21.Square numbers 1, 4, 9, 16, 25 — each new picture adds one more row and one more column. The next picture is 6 × 6 = 36.
Q2.
Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes?
Answer
Because of the shape the dots make when you arrange them neatly:
Triangular numbers — that many dots can be packed into a perfect triangle: rows of 1, 2, 3, 4, … dots stacked one below the other. So 1 + 2 + 3 = 6 dots make a triangle of 3 rows, and 6 is a triangular number.
Square numbers — that many dots can be packed into a perfect square grid with the same number of rows and columns. 4 = 2 × 2, 9 = 3 × 3, 25 = 5 × 5.
Cubes — that many dots (or small unit cubes) can be packed into a solid cube with equal length, breadth and height. 8 = 2 × 2 × 2, 27 = 3 × 3 × 3, 125 = 5 × 5 × 5.
The idea: a number gets its name from the shape it can be arranged into. The same number can therefore have more than one name — see the very next question.
Q3.
You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this!
Answer
Yes — 36 wears two hats at once:
As a triangle: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 (8 rows)
As a square: 6 × 6 = 36 (6 rows of 6)
The same 36 dots — arranged as a triangle of 8 rows, and as a 6 × 6 square.
This shows that a number can play different roles depending on the context. Other numbers can also be shown in more than one way:
1 is a triangular number, a square number and a cube.
16 is a square (4 × 4) and also a rectangle (2 × 8).
64 is both a square (8 × 8) and a cube (4 × 4 × 4).
Did you know? After 1 and 36, the next number that is both triangular and square is 1225 (= 35 × 35, and also 1 + 2 + 3 + … + 49). Such numbers are very rare.
Q4.
What would you call the following sequence of numbers? 1, 7, 19, 37 … That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?
Answer
These are hexagonal numbers — the dots form a hexagon (a six-sided figure) with one dot in the centre and rings of dots around it.
Hexagonal numbers 1, 7, 19, 37. Each new picture adds one more ring around the outside; the rings hold 6, 12 and 18 dots.
Look at how much is added each time:
1 +6→ 7 +12→ 19 +18→ 37 +24→ 61
The additions are 6, 12, 18, 24, … (multiples of 6). So the next ring holds 24 dots and
37 + 24 = 61
Answer: the next hexagonal number is 61. The sequence continues 1, 7, 19, 37, 61, 91, 127, …
Q5.
Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Answer
Yes. The trick is to show that each picture is made of copies of the picture before it.
Powers of 2 — every picture is two copies of the previous one, so the number of dots doubles: 1, 2, 4, 8, 16, …
Powers of 2: 1, 2, 4, 8, 16. Each step is simply the previous picture, taken twice.
Powers of 3 — every picture is three copies of the previous one, so the number of dots triples: 1, 3, 9, 27, …
Powers of 3: 1, 3, 9, 27. A single dot; three dots in a triangle; three of those triangles; three of those groups again.
Why this picture is useful: it shows at a glance why the numbers grow so fast — at every step the whole picture is copied 2 (or 3) times. This is called doubling and tripling.