NCERT Solutions Ganita Prakash Chapter 1 Figure it Out — Relations among Number Sequences
Book page 8 & 9 Updated on2026-09-05
Q1.
Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …, gives square numbers?
Answer
Yes. Take a square grid of dots and colour it along the slanting diagonals instead of along L-shapes.
A 6 × 6 square cut along its diagonals. The diagonals contain 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 dots — counting up and then down.
Count the dots on each slanting line of a 6 × 6 square. Starting from the top-left corner they contain
1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 dots
Every dot of the square lies on exactly one of these diagonals, so
The same picture can be drawn for any size of square, which explains why adding counting numbers up and then down always gives a square number.
Q2.
By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
Answer
In the picture above, the largest number in the middle tells us the side of the square. Here the middle number is 100, so imagine a 100 × 100 square of dots.
Why: going “up and down” to a height of n uses n ones on the way up and n − 1 ones on the way down, that is n + (n − 1) = 2n − 1 ones in all — and 2n − 1 is exactly the nth odd number.
Q4.
Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
We get 1, 3, 6, 10, 15, … — the triangular numbers.
Rows of 1, 2, 3, 4 and 5 dots stacked one below the other make a triangle of 15 dots. That is why the sums are called triangular numbers.
The picture explains it immediately: putting a row of 1 dot, then 2 dots, then 3 dots, … one below the other builds the triangle itself. So the running totals of the counting numbers are exactly the triangular numbers.
Q5.
What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … Which sequence do you get? Why? Can you explain it with a picture?
A 5 × 5 square split by its diagonal: the green part is a triangle of 15 dots, the orange part a triangle of 10 dots. Together 10 + 15 = 25.
Why it happens: cut a square of dots along its main diagonal. One piece (including the diagonal) is a triangle of dots, the other piece is the previous, slightly smaller triangle. So two consecutive triangular numbers always fit together to fill a square exactly.
Q6.
What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, …? Now add 1 to each of these numbers — what numbers do you get? Why does this happen?
We get 2, 4, 8, 16, 32, … — the powers of 2 again!
Why it happens: think of the doubling picture. Take the pile of 1 + 2 + 4 + 8 dots and add one more single dot. Now 1 + 1 = 2, and that 2 joins the next 2 to make 4, and that 4 joins the next 4 to make 8, and so on — everything collapses into a single pile of 16. In short:
1 + 2 + 4 + … + 2n−1 = 2n − 1, so adding 1 gives exactly 2n.
Q7.
What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?
We get 7, 19, 37, 61, 91, … — the hexagonal numbers (from question 4 of page 5).
The hexagonal number 37 = 6 × 6 + 1. The black dot is the centre; the six coloured wedges around it are six identical triangles of 6 dots each.
Why it happens: a hexagon is made of six identical triangles meeting at one central point. Remove the centre dot and the remaining dots split into 6 equal triangular groups. So every hexagonal number is “6 × a triangular number, plus the 1 dot in the middle”.
Q8.
What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, …? Which sequence do you get? Can you explain it using a picture of a cube?
Why it happens (the cube picture): build a cube out of small unit cubes, one shell at a time.
Start with 1 small cube — that is a 1 × 1 × 1 cube.
To turn it into a 2 × 2 × 2 cube you must add 7 more small cubes (8 − 1 = 7).
To turn that into a 3 × 3 × 3 cube you add 19 more (27 − 8 = 19).
To reach 4 × 4 × 4 you add 37 more (64 − 27 = 37).
The number of cubes in each new shell is exactly 1, 7, 19, 37, … — the hexagonal numbers. So adding hexagonal numbers rebuilds the cube layer by layer, and the totals are the cubes.
Check it yourself: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37, 125 − 64 = 61. The differences of consecutive cubes are the hexagonal numbers.
Q9.
Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
Answer
Here are some patterns you can discover — try to find more of your own.
Adding even numbers: 2, 2 + 4 = 6, 2 + 4 + 6 = 12, 2 + 4 + 6 + 8 = 20 → 2, 6, 12, 20, 30 … These are exactly double the triangular numbers, and each one is a rectangle of dots: 1 × 2, 2 × 3, 3 × 4, 4 × 5, …
Differences of squares: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9 → the odd numbers. (This is the same L-shape picture as on page 7, read backwards.)
Virahānka numbers: add the first few — 1 + 2 + 3 + 5 = 11, which is 2 less than 13, the next-but-one Virahānka number. Try it again: 1 + 2 + 3 + 5 + 8 = 19 = 21 − 2. The pattern holds!
Powers of 2 and odd/even: every power of 2 after 1 is even, and no power of 3 is ever even.
Try This: Choose any two sequences from Table 1 and add them term by term. For example odd + even: 1 + 2 = 3, 3 + 4 = 7, 5 + 6 = 11 → 3, 7, 11, 15 … a new sequence that goes up by 4 each time. Explain why.