Chapter 1 exercise answers: Patterns in Mathematics

Open NCERT p. 2One way to think about it

1. Mathematics is not only the sums done in a maths period.
2. Cooking uses it: scaling a recipe up or down means multiplying or dividing every quantity.
3. Shopping uses it: comparing prices and working out change both need arithmetic.
4. Sport uses it: keeping score, timing a race, or working out a run rate.
5. Travel uses it: reading a distance, a speed and working out how long a journey will take.
6. Any everyday task that needs counting, measuring, comparing or planning ahead is an example.

In shortEveryday activities such as cooking (scaling a recipe), shopping (comparing prices, working out change), sport (keeping score) and travel (working out journey time) all use mathematics; any activity you do that needs counting, measuring or comparing is a valid example.

Watch this explained “Mathematics in an ordinary day”, 6:48 into Mathematics is the search for patterns *and* for why they hold · हिंदी में देखें

Open NCERT p. 2One way to think about it

1. People noticed patterns in how stars, planets and moons move across the sky.
2. Asking why those patterns happen led to the theory of gravitation.
3. That theory then let humans launch satellites and send rockets to the Moon and Mars.
4. Mathematics is also used to run economies and elections, since both depend on counting and fair calculation.
5. It is used to work out the loads and measurements needed to build bridges, houses and other structures safely.
6. It is behind the engineering of TVs, phones, computers, bicycles, trains, cars and planes.
7. In each case, a pattern noticed for one reason turned into a reason that was useful somewhere completely different.

In shortMathematics has propelled humanity forward because a reason discovered in one place (such as the theory of gravitation, found by studying the sky) can be carried into a new situation (such as launching satellites); this is also how it underlies building bridges and complex structures, running economies fairly, and engineering machines like cars, planes and computers.

Watch this explained “The sky, gravitation, and satellites”, 4:17 into Mathematics is the search for patterns *and* for why they hold · हिंदी में देखें

Open NCERT p. 3One way to think about it

1. Sequence	Rule for the next term
   All 1's: 1, 1, 1, 1, 1, …	Every term is 1; nothing changes.
   Counting numbers: 1, 2, 3, 4, 5, …	Add 1 to the term before it.
   Odd numbers: 1, 3, 5, 7, 9, …	Add 2 to the term before it, starting from 1.
   Even numbers: 2, 4, 6, 8, 10, …	Add 2 to the term before it, starting from 2.
   Triangular numbers: 1, 3, 6, 10, 15, …	Add the next counting number (add 2, then 3, then 4, …).
   Squares: 1, 4, 9, 16, 25, …	The nth term is n × n.
   Cubes: 1, 8, 27, 64, 125, …	The nth term is n × n × n.
   Virahānka numbers: 1, 2, 3, 5, 8, 13, 21, …	Add the two terms before it.
   Powers of 2: 1, 2, 4, 8, 16, …	Multiply the term before it by 2.
   Powers of 3: 1, 3, 9, 27, 81, …	Multiply the term before it by 3.

In shortEach of the ten sequences in Table 1 has its own simple growing rule, shown in the table above — some add a fixed or growing amount, and some multiply by a fixed amount.

Watch this explained “Ten rows, ten machines”, 2:25 into Every number sequence is a rule, not a list · हिंदी में देखें

Open NCERT p. 3One way to think about it

1. Sequence	Next three terms	Rule, in words
   1, 1, 1, 1, 1, …	1, 1, 1	Every term is 1.
   1, 2, 3, 4, 5, 6, 7, …	8, 9, 10	Add 1 each time.
   1, 3, 5, 7, 9, 11, 13, …	15, 17, 19	Add 2 each time, starting from 1.
   2, 4, 6, 8, 10, 12, 14, …	16, 18, 20	Add 2 each time, starting from 2.
   1, 3, 6, 10, 15, 21, 28, …	36, 45, 55	Add one more than was added last time (add 2, 3, 4, 5, 6, 7, 8, …).
   1, 4, 9, 16, 25, 36, 49, …	64, 81, 100	The nth term is n × n.
   1, 8, 27, 64, 125, 216, …	343, 512, 729	The nth term is n × n × n.
   1, 2, 3, 5, 8, 13, 21, …	34, 55, 89	Add the two previous terms to get the next one.
   1, 2, 4, 8, 16, 32, 64, …	128, 256, 512	Multiply the previous term by 2.
   1, 3, 9, 27, 81, 243, 729, …	2187, 6561, 19683	Multiply the previous term by 3.

In shortThe next three terms and the rule for each of the ten sequences are shown in the table above.

Watch this explained “Machines with a fixed step”, 3:13 into Every number sequence is a rule, not a list · हिंदी में देखें

Open NCERT p. 5One way to think about it

1. All 1's: the next picture is again a single dot (the 6th term is 1).
2. Counting numbers: a single row of 6 dots (6).
3. Odd numbers: a row of 5 dots above a row of 6 dots, so exactly one dot sticks out (5 + 6 = 11).
4. Even numbers: two equal rows of 6 dots (6 + 6 = 12).
5. Triangular numbers: add a row of 6 dots under the triangle of 15, giving rows of 1, 2, 3, 4, 5, 6 dots (21).
6. Squares: a 6-by-6 square of dots (6 × 6 = 36).
7. Cubes: a solid cube with 6 little cubes along each edge (6 × 6 × 6 = 216).

In shortThe next (6th) pictures are: 1 dot; a row of 6 dots; a row of 5 above a row of 6 (11 dots); two rows of 6 (12 dots); a triangle with rows of 1 to 6 (21 dots); a 6-by-6 square (36 dots); and a cube with 6 little cubes along each edge (216 cubes).

Watch this explained “The task is the drawing”, 6:16 into Drawing a sequence makes its rule visible · हिंदी में देखें

Open NCERT p. 5One way to think about it

1. A name in this chapter comes from the picture the numbers make, not from the numbers alone.
2. 1, 3, 6, 10, 15, … dots can each be arranged as a full triangle (a row of 1, then 2, then 3, …), so they are called triangular numbers.
3. 1, 4, 9, 16, 25, … dots can each be arranged as a full square grid (1×1, 2×2, 3×3, …), so they are called square numbers.
4. 1, 8, 27, 64, 125, … unit cubes can each be built into a full solid cube (1×1×1, 2×2×2, 3×3×3, …), so they are called cubes.

In shortEach name describes the shape the dots or cubes make: triangular numbers can be arranged in a triangle, square numbers in a square, and cubes in a solid cube.

Watch this explained “The triangular numbers earn their name”, 3:57 into Drawing a sequence makes its rule visible · हिंदी में देखें

Open NCERT p. 5One way to think about it

1. As a square: draw a 6-by-6 grid of dots — 6 rows of 6 dots — which is 36 dots in total.
2. As a triangle: draw rows of 1, 2, 3, 4, 5, 6, 7 and 8 dots stacked on top of each other — these rows also add up to 36 dots.
3. Both pictures use exactly 36 dots, just arranged differently, which is why 36 is both a square number and a triangular number.
4. Other numbers can also be tried: for example, 1 is both the 1st triangular number and the 1st square number, since a single dot is trivially both shapes.

In short36 dots can be arranged as a 6×6 square, and separately as a triangle with rows of 1 to 8 dots (1+2+3+4+5+6+7+8 = 36); the same 36 dots simply play two different roles depending on how they are arranged.

Watch this explained “36, twice — one number, two buildings”, 7:15 into Drawing a sequence makes its rule visible · हिंदी में देखें

Open NCERT p. 5Matches NCERT’s answer

1. The sequence so far is 1, 7, 19, 37, ….
2. Find how much is added each time: 7 − 1 = 6, 19 − 7 = 12, 37 − 19 = 18.
3. The amount added is itself growing by 6 each time: 6, 12, 18, so the next amount added is 18 + 6 = 24.
4. Add this to the last term: 37 + 24 = 61.

Answer61

Watch this explained “A sequence named from its picture”, 8:23 into Drawing a sequence makes its rule visible · हिंदी में देखें

Open NCERT p. 5One way to think about it

1. There is more than one good way to picture these sequences; here is one for each.
2. Powers of 2 (the book's own picture on page 6): a dot (1); copy it and join the two dots with a line (2); copy that line and join matching ends to get a square (4 corners); copy the square and join matching corners to get a cube (8 corners); two cubes joined corner to corner give 16. Each picture is 2 copies of the one before, so the count doubles.
3. Powers of 3: a dot (1); a row of 3 dots (3); 3 such rows make a 3-by-3 square of dots (9); 3 such squares stacked as layers make a 3 × 3 × 3 cube of dots (27); 3 such cubes placed side by side make 81 dots. Each picture is 3 copies of the one before, so the count is multiplied by 3.

In shortOne way (others are also correct): build each picture from copies of the one before — 2 copies for powers of 2 (dot, line, square, cube, two joined cubes: 1, 2, 4, 8, 16 corners) and 3 copies for powers of 3 (dot, row of 3, 3-by-3 square, 3 × 3 × 3 cube, three such cubes: 1, 3, 9, 27, 81 dots).

Watch this explained “The powers of 2 as growing shapes”, 10:39 into Drawing a sequence makes its rule visible · हिंदी में देखें

Open NCERT p. 8One way to think about it

1. Take a square grid of dots, say 6 by 6 (36 dots).
2. Cut it along its diagonals instead of in corner-bands: the top-left corner starts one diagonal.
3. The diagonal lines hold 1, 2, 3, 4, 5, 6 dots going one way to the middle, then 5, 4, 3, 2, 1 dots coming back down the other way.
4. Adding these diagonal counts gives exactly 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1, which is 36 — the same square number.
5. The same cut works for a square grid of any size, so counting numbers added up and back down always give a square number.

In shortCutting a square of dots along its diagonals (instead of in corner-bands) splits it into rows of 1, 2, 3, …, up to the side length and then back down to 1; adding these diagonal counts reproduces the square total, which is why counting numbers added up and down give square numbers.

Watch this explained “The same square, sliced the other way”, 7:16 into Why adding the odd numbers gives the squares · हिंदी में देखें

Open NCERT p. 8Matches NCERT’s answer

1. This is the up-and-down pattern, which always gives a square number equal to (the peak number) squared.
2. Here the peak number is 100.
3. So the value is 100 × 100.

Answer10000

Watch this explained “The same square, sliced the other way”, 7:16 into Why adding the odd numbers gives the squares · हिंदी में देखें

Open NCERT p. 8Checked by computer

1. The All 1's sequence is 1, 1, 1, 1, 1, ….
2. Adding it up (running total): 1, then 1+1=2, then 1+1+1=3, then 1+1+1+1=4, … — this gives the counting numbers.
3. Adding it up and down (like 1, 1+2+1, 1+2+3+2+1, … but using 1's): row k is k ones going up plus (k−1) ones coming back down, i.e. k + (k−1) = 2k−1.
4. That gives 1, 3, 5, 7, 9, … — the odd numbers.

AnswerAdding the All 1's sequence up gives the counting numbers (1, 2, 3, 4, 5, …); adding it up and down gives the odd numbers (1, 3, 5, 7, 9, …).

Watch this explained “Warm-up — All 1's becomes the counting numbers”, 1:30 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 8One way to think about it

1. Counting numbers: 1, 2, 3, 4, 5, ….
2. Adding them up as a running total: 1, then 1+2=3, then 1+2+3=6, then 1+2+3+4=10, then 1+2+3+4+5=15, ….
3. This gives 1, 3, 6, 10, 15, … — the triangular numbers.
4. Picture: draw a row of 1 dot, then under it a row of 2, then a row of 3, and so on; each new triangle picture is made by adding one more row, which is exactly one more counting number.

In shortAdding the counting numbers up gives the triangular numbers (1, 3, 6, 10, 15, …); a small picture is simply the triangle-of-dots picture, since adding one more row (of the next counting number) each time is exactly how a triangular number is built.

Watch this explained “Counting numbers become the triangular numbers”, 2:23 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 8One way to think about it

1. 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, ….
2. These are 4, 9, 16, 25, … — the square numbers.
3. Picture: take a triangle of dots for a number, and another triangle of dots for the next triangular number; turn the second triangle upside down and fit it against the first.
4. The two triangles fit together exactly, with no gaps or overlaps, to form a square, which is why two consecutive triangular numbers always add up to a square number.

In shortAdding two consecutive triangular numbers always gives a square number (1+3=4, 3+6=9, 6+10=16, 10+15=25, …); a triangle of dots and the next, upside down, fit together perfectly to make a square.

Watch this explained “Two triangles make a square”, 3:52 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 8One way to think about it

1. Running totals: 1 = 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15, 1 + 2 + 4 + 8 + 16 = 31.
2. Add 1 to each: 2, 4, 8, 16, 32 — these are the powers of 2 (starting from 2).
3. Why: start with 1 + 1 = 2. Adding the next power, 2, gives 2 + 2 = 4. Adding the next power, 4, gives 4 + 4 = 8. Then 8 + 8 = 16, and so on.
4. At every step, the total plus 1 is exactly the next power of 2 to be added, so adding that power doubles it. Starting from 2 and doubling each time gives the powers of 2.

In shortThe totals are 1, 3, 7, 15, 31, …; adding 1 to each gives 2, 4, 8, 16, 32, … — the powers of 2. This happens because the total plus 1 always equals the next power to be added, so each step doubles it: 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16.

Watch this explained “Powers of 2, one short every time”, 4:59 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 9One way to think about it

1. Triangular numbers: 1, 3, 6, 10, ….
2. 6 × 1 + 1 = 7, 6 × 3 + 1 = 19, 6 × 6 + 1 = 37, 6 × 10 + 1 = 61.
3. These are 7, 19, 37, 61, … — the hexagonal numbers (after the first one, 1).
4. Picture: a hexagonal-number picture is a centre dot surrounded by 6 triangles of dots; each of those 6 triangles is a triangular number, so 6 times a triangular number plus the 1 centre dot gives the hexagonal number.

In shortMultiplying triangular numbers by 6 and adding 1 gives 7, 19, 37, 61, … — the hexagonal numbers; a hexagonal picture is 1 centre dot surrounded by 6 identical triangles of dots, which is exactly 6 times a triangular number plus 1.

Watch this explained “Six triangles round a centre dot”, 6:07 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 9One way to think about it

1. Running totals: 1 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64.
2. These are 1, 8, 27, 64, … — the cubes (1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, 4 × 4 × 4).
3. Picture: look at the book's 4 × 4 × 4 cube from one corner. You see 1 corner cube (yellow), 3 edges of 3 cubes each (9, red) and 3 faces of 3 × 3 cubes each (27, blue): 1 + 9 + 27 = 37 cubes, the 4th hexagonal number. From that corner this layer looks just like the hexagon of 37 dots beside it.
4. Remove that layer and a 3 × 3 × 3 cube is left; its visible layer is 1 + 6 + 12 = 19 cubes. Removing that leaves a 2 × 2 × 2 cube whose visible layer is 1 + 3 + 3 = 7 cubes, and then a single cube, 1.
5. So the 4 × 4 × 4 cube is made of layers of 1, 7, 19 and 37 cubes, and 1 + 7 + 19 + 37 = 64. The same peeling works for a cube of any size, which is why adding up hexagonal numbers gives the cubes.

In shortAdding up hexagonal numbers gives the cubes 1, 8, 27, 64, …, because a cube can be peeled, from one corner, into layers of 1, 7, 19, 37, … little cubes, and each layer looks like a hexagon when seen from that corner.

Watch this explained “Reading the book's cube picture”, 8:13 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 9One way to think about it

1. There are many correct answers; here is one.
2. Add up the even numbers: 2, 2 + 4 = 6, 2 + 4 + 6 = 12, 2 + 4 + 6 + 8 = 20, 2 + 4 + 6 + 8 + 10 = 30.
3. Compare with the triangular numbers 1, 3, 6, 10, 15: each total is exactly twice a triangular number (2 × 1, 2 × 3, 2 × 6, 2 × 10, 2 × 15).
4. Why: Table 2 draws each even number as two equal rows of dots. Adding up 2, 4, 6, 8, 10 therefore stacks two copies of the rows 1, 2, 3, 4, 5 — that is, two triangles of 15 dots.
5. The two triangles also fit together into a rectangle of 5 rows of 6 dots, so the total is 5 × 6 = 30.

In shortOne relation (many are possible): adding up the even numbers gives 2, 6, 12, 20, 30, …, which is twice the triangular numbers 1, 3, 6, 10, 15, …, because each even number is two equal rows of dots, so the running total is two copies of a triangle of dots.

Watch this explained “Find your own relation”, 10:51 into Sequences that turn out to be the same sequence in disguise · हिंदी में देखें

Open NCERT p. 11One way to think about it

1. Shape sequence	Rule for the next shape
   Regular polygons	Add one more side, keeping all sides equal and all corners alike.
   Complete graphs	Add one more dot, and join it by a line to every dot already there.
   Stacked squares	Make each side one little square longer, and fill it in with little squares.
   Stacked triangles	Add one more row of little triangles along the bottom.
   Koch snowflake	Replace every straight segment by a 'speed bump', so each segment becomes 4 smaller segments.

In shortEach of the five shape sequences in Table 3 grows by its own simple rule — see the table above.

Watch this explained “Row 1: the regular polygons”, 2:06 into Shapes come in sequences too, with rules of their own · हिंदी में देखें

Open NCERT p. 11One way to think about it

1. Regular polygons: yes — the next shape is a regular 11-sided polygon. Rule: add one more side, keeping all sides equal and all corners alike.
2. Complete graphs: yes — the next shape is K7: 7 dots with every pair joined (the new dot is joined to the 6 dots already there). Rule: add one dot and join it to every other dot.
3. Stacked squares: yes — a 6-by-6 block of little squares. Rule: make each side one little square longer and fill it in.
4. Stacked triangles: yes — a big triangle with 6 rows of little triangles. Rule: add one more row of little triangles along the bottom.
5. Koch snowflake: the rule still says exactly what to do — replace each of the 768 segments of the last shape with a speed bump, giving 3072 segments — but the bumps are now too tiny to draw by hand. The next shape exists; only the pencil cannot show it.

In shortThe next shape can be drawn for the regular polygons (11 sides), complete graphs (K7), stacked squares (6 by 6) and stacked triangles (6 rows). For the Koch snowflake the rule still gives a next shape (3072 segments), but its bumps are too small to draw by hand.

Watch this explained “Four yes, and one awkward”, 8:02 into Shapes come in sequences too, with rules of their own · हिंदी में देखें

Open NCERT p. 11One way to think about it

1. Sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10 — the counting numbers starting at 3.
2. Corners: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10 — exactly the same sequence.
3. This happens because walking around any closed straight-sided shape, you arrive at exactly one corner after walking along exactly one side, and you end up back where you started.
4. So a polygon must always have the same number of corners as it has sides.

In shortBoth counting the sides and counting the corners of the Regular Polygons gives the same sequence, 3, 4, 5, 6, 7, 8, 9, 10, …, because walking around a closed shape passes exactly one corner for every one side, so the two counts always match.

Watch this explained “Two counts, one answer — and the reason”, 3:36 into Counting the parts of a shape sequence produces a number sequence · हिंदी में देखें

Open NCERT p. 11One way to think about it

1. K2 has 1 line, K3 has 3 lines, K4 has 6 lines, K5 has 10 lines, K6 has 15 lines.
2. This gives 1, 3, 6, 10, 15, … — the triangular numbers.
3. Going from one complete graph to the next, one new dot is added, and it must be joined by a new line to every dot already there.
4. So the number of new lines added each time is 1, 2, 3, 4, 5, … — exactly how the triangular numbers are built up.

In shortCounting the lines in the Complete Graphs gives the triangular numbers (1, 3, 6, 10, 15, …), because each new dot added must be joined to every earlier dot, adding one more line than was added the time before.

Watch this explained “Complete graphs: count the lines”, 4:43 into Counting the parts of a shape sequence produces a number sequence · हिंदी में देखें

Open NCERT p. 12One way to think about it

1. The shapes have 1, 4, 9, 16, 25 little squares.
2. This gives 1, 4, 9, 16, 25, … — the square numbers.
3. Each big shape of side n little squares is made of n rows, with n little squares in each row.
4. n rows of n little squares is n × n little squares in total, which is exactly the square numbers.

In shortCounting the little squares in the Stacked Squares gives the square numbers (1, 4, 9, 16, 25, …), because a shape with side n little squares is simply n rows of n little squares, i.e. n × n.

Watch this explained “Stacked squares: n rows of n”, 6:25 into Counting the parts of a shape sequence produces a number sequence · हिंदी में देखें

Open NCERT p. 12One way to think about it

1. Counting carefully, including the small upside-down triangles, the shapes have 1, 4, 9, 16, 25 little triangles.
2. This gives 1, 4, 9, 16, 25, … — the square numbers, the same sequence as the Stacked Squares.
3. Row by row from the top, a shape with n rows has 1, 3, 5, 7, …, (2n−1) little triangles in each successive row (upward- and downward-pointing triangles together).
4. Adding 1 + 3 + 5 + ⋯ + (2n−1) for n rows always gives n × n, since adding up the first n odd numbers gives a square number.

In shortCounting the little triangles in the Stacked Triangles also gives the square numbers (1, 4, 9, 16, 25, …); row n from the top has (2n−1) little triangles, and adding up the first n odd numbers (1, 3, 5, …) always gives n².

Watch this explained “Stacked triangles: 1, 3, 5, 7 …”, 7:09 into Counting the parts of a shape sequence produces a number sequence · हिंदी में देखें

Open NCERT p. 12Matches NCERT’s answer

1. The first shape is a triangle, which has 3 straight sides, so it has 3 line segments.
2. Every time the picture is redrawn, each existing straight segment is replaced by a 'speed bump' made of 4 segments, so the total is multiplied by 4 each time.
3. 3, 3×4=12, 12×4=48, 48×4=192, 192×4=768, ….
4. So the number sequence is 3, 12, 48, 192, 768, …, which is 3 times the powers of 4 (3×4⁰, 3×4¹, 3×4², …).

Answer3, 12, 48, 192, 768, …

Watch this explained “Koch, and a sequence Table 1 never had”, 9:27 into Counting the parts of a shape sequence produces a number sequence · हिंदी में देखें