Patterns In MathematicsClass 6 Mathematics NCERT Solutions
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Q1Section 1.1 Figure it Out
Can you think of other examples where mathematics helps us in our everyday lives?
Solution
Mathematics is used in many aspects of our everyday lives. Here are some examples:
- Shopping: Calculating total bills, discounts, and taxes involves basic arithmetic.
- Cooking and Baking: Measuring ingredients accurately requires understanding fractions and ratios.
- Time Management: Reading clocks, creating schedules, and calculating travel time all use mathematical concepts.
- Budgeting: Managing personal or household finances involves addition, subtraction, and percentages to track income and expenses.
- Construction and DIY Projects: Measuring lengths, calculating areas, and cutting materials at correct angles are applications of geometry.
- Sports: Understanding game scores, player statistics, and angles for a shot involves numbers and geometry.
- Navigation: Using maps or GPS involves concepts of distance, scale, and coordinates.
Q2Section 1.1 Figure it Out
How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)
Solution
Mathematics has been fundamental to the advancement of human civilization in countless ways:
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Science and Technology: The laws of physics, chemistry, and biology are expressed in mathematical equations. This allows us to carry out scientific experiments, understand the universe, and develop new technologies like computers, mobile phones, and medical equipment.
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Engineering and Architecture: Building safe and efficient structures like bridges, skyscrapers, and dams requires precise calculations using principles from geometry, calculus, and physics. Similarly, designing vehicles like cars, trains, and airplanes relies heavily on mathematics.
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Economy and Finance: Economics uses mathematical models to understand market trends, set policies, and manage investments. Banking, insurance, and stock markets are all built on mathematical principles.
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Information and Communication: The digital world runs on mathematics. Computers use binary logic, and data is secured with encryption algorithms based on number theory. The internet, television, and phone networks all use mathematical codes to transmit information.
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Democracy and Society: Statistics, a branch of mathematics, is used to conduct censuses, analyze survey data, and ensure fair representation in democratic processes.
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Timekeeping: The development of accurate calendars and clocks, essential for agriculture, navigation, and modern life, was a major mathematical achievement.
Q1Section 1.2 Figure it Out
Can you recognise the pattern in each of the sequences in Table 1?
Solution
Yes, each sequence in Table 1 follows a specific rule or pattern:
- 1, 1, 1, 1, 1, 1, 1, ... (All 1's): Every number in the sequence is 1.
- 1, 2, 3, 4, 5, 6, 7, ... (Counting numbers): The sequence starts with 1, and each subsequent number is obtained by adding 1 to the previous number.
- 1, 3, 5, 7, 9, 11, 13, ... (Odd numbers): The sequence starts with 1, and each subsequent number is obtained by adding 2 to the previous number.
- 2, 4, 6, 8, 10, 12, 14, ... (Even numbers): The sequence starts with 2, and each subsequent number is obtained by adding 2 to the previous number.
- 1, 3, 6, 10, 15, 21, 28, ... (Triangular numbers): The sequence starts with 1. To get the next number, you add the next counting number. (, , , , and so on).
- 1, 4, 9, 16, 25, 36, 49, ... (Squares): Each number is the square of its position in the sequence (, and so on).
- 1, 8, 27, 64, 125, 216, ... (Cubes): Each number is the cube of its position in the sequence (, and so on).
- 1, 2, 3, 5, 8, 13, 21, ... (Virahānka numbers): After the first two numbers, each subsequent number is the sum of the two preceding numbers (, and so on).
- 1, 2, 4, 8, 16, 32, 64, ... (Powers of 2): The sequence starts with 1, and each subsequent number is obtained by multiplying the previous number by 2.
- 1, 3, 9, 27, 81, 243, 729, ... (Powers of 3): The sequence starts with 1, and each subsequent number is obtained by multiplying the previous number by 3.
Q2Section 1.2 Figure it Out
Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
Solution
Here are the sequences with their next three numbers and the rule for each:
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(All 1's): 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... Rule: Every number in the sequence is 1.
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(Counting numbers): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... Rule: Start with 1 and add 1 to the previous number to get the next number.
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(Odd numbers): 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ... Rule: Start with 1 and add 2 to the previous number to get the next number.
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(Even numbers): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... Rule: Start with 2 and add 2 to the previous number to get the next number.
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(Triangular numbers): 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... Rule: To get the next number, add the next consecutive counting number to the previous number ().
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(Squares): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... Rule: The numbers are the squares of the counting numbers ().
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(Cubes): 1, 8, 27, 64, 125, 216, 343, 512, 729, ... Rule: The numbers are the cubes of the counting numbers ().
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(Virahānka numbers): 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... Rule: Add the two previous numbers to get the next number ().
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(Powers of 2): 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Rule: Start with 1 and multiply the previous number by 2 to get the next number.
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(Powers of 3): 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ... Rule: Start with 1 and multiply the previous number by 3 to get the next number.
Q1Section 1.3 Figure it Out
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
Solution
The next picture for each sequence is described below:
- All 1's: The next picture is a single dot, just like all the others.
- Counting numbers: The next picture is a single row of 8 dots.
- Odd numbers: The next picture is an L-shaped arrangement of 15 dots added around a square of dots.
- Even numbers: The next picture is a rectangle of dots with 2 rows and 8 columns, for a total of 16 dots.
- Triangular numbers: The next picture is a triangle of dots with 8 dots in its base row. The total number of dots will be .
- Squares: The next picture is a square grid of dots with 8 rows and 8 columns, for a total of dots.
- Cubes: The next picture is a representation of a cube made of dots that is 4 dots long, 4 dots wide, and 4 dots high, for a total of dots.
Q2Section 1.3 Figure it Out
Why are called triangular numbers? Why are called square numbers or squares? Why are called cubes?
Solution
These sequences are named after the geometric shapes their numbers can form:
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Triangular numbers: The numbers in the sequence can be arranged to form equilateral triangles. For example, 3 dots can form a triangle with 2 dots on each side, 6 dots can form a triangle with 3 dots on each side, and so on.
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Square numbers: The numbers in the sequence can be arranged to form squares. For example, 4 dots can form a square, 9 dots can form a square, and so on.
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Cubes: The numbers in the sequence represent the number of smaller, equal-sized cubes required to build a larger cube. For example, 8 smaller cubes can form a larger cube, 27 smaller cubes can form a larger cube, and so on.
Q3Section 1.3 Figure it Out
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!
Solution
To illustrate that 36 is both a square and a triangular number, you can draw two different arrangements of 36 dots:
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As a square: Draw a grid of dots with 6 rows and 6 columns. This arrangement will contain dots and will form a perfect square.
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As a triangle: Draw a triangle made of dots. The first row will have 1 dot, the second will have 2, the third will have 3, and so on, until the last row, which will have 8 dots. The total number of dots will be . This arrangement will form a perfect equilateral triangle.
Q4Section 1.3 Figure it Out
What would you call the following sequence of numbers? 1, 7, 19, 37, 61, ... That's right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?
Solution
To find the next number in the sequence:
First, let's find the pattern by looking at the differences between consecutive terms:
The differences are multiples of 6: . The next difference in this pattern should be .
To find the next number in the original sequence, we add this difference to the last term:
Final Answer: The next number in the sequence is 91.
Q5Section 1.3 Figure it Out
Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Solution
Yes, there are several pictorial ways to visualize these sequences.
Powers of 2 (1, 2, 4, 8, ...):
One way is to use a branching diagram.
- Start with a single point (representing 1).
- From that point, draw two branches, leading to two new points (representing 2).
- From each of these two points, draw two more branches, leading to four new points (representing 4).
- Continue this process. At each step, the number of new points doubles, representing the next power of two.
Powers of 3 (1, 3, 9, 27, ...):
A similar branching diagram can be used.
- Start with a single point (representing 1).
- From that point, draw three branches, leading to three new points (representing 3).
- From each of these three points, draw three more branches, leading to nine new points (representing 9).
- At each step, the number of new points triples, representing the next power of three.
Q1Section 1.4 Figure it Out
Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., , gives square numbers?
Solution
Yes, a pictorial explanation can be found by looking at the diagonals of a square grid of dots.
Consider a square grid of dots, for example, a square which has 16 dots.
- Start at one corner. There is 1 dot.
- The next diagonal line of dots parallel to the main diagonal has 2 dots.
- The next diagonal (the main one) has 3 dots.
- The longest diagonal has 4 dots.
- After the longest diagonal, the number of dots in the diagonals decreases: 3 dots, then 2 dots, and finally 1 dot in the opposite corner.
By adding the number of dots in each of these diagonal slices, we get the total number of dots in the square: , which is . This shows that adding counting numbers up to a peak and then back down gives a square number.
Q2Section 1.4 Figure it Out
By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of ?
Solution
Pattern: The sum is equal to .
To Find: The value of .
Solution:
In this sum, the peak number 'n' is 100. This sum represents the total number of dots in a square grid.
Therefore, the value of the sum is .
Final Answer: The value of the sum is 10,000.
Q3Section 1.4 Figure it Out
Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down?
Solution
1. Adding the All 1's sequence up:
- 1st term:
- 2nd term:
- 3rd term:
- 4th term: The resulting sequence is 1, 2, 3, 4, ..., which is the Counting numbers sequence.
2. Adding the All 1's sequence up and down:
- 1st term:
- 2nd term:
- 3rd term:
- 4th term: The resulting sequence is 1, 3, 5, 7, ..., which is the Odd numbers sequence.
Q4Section 1.4 Figure it Out
Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?
Solution
1. The sequence:
When we add the counting numbers up, we get:
- 1st term:
- 2nd term:
- 3rd term:
- 4th term: The resulting sequence is 1, 3, 6, 10, ..., which is the Triangular numbers sequence.
2. Pictorial Explanation:
We can visualize this by building triangles with dots.
- Start with 1 dot.
- To get the next number (3), add a new row of 2 dots below the first dot. This forms a small triangle.
- To get the next number (6), add a new row of 3 dots below the triangle of 3 dots. Each step involves adding the next counting number as a new row to the base of the triangle, thus forming the next triangular number.
Q5Section 1.4 Figure it Out
What happens when you add up pairs of consecutive triangular numbers? That is, take Which sequence do you get? Why? Can you explain it with a picture?
Solution
1. The sequence:
Let's calculate the sums:
- The resulting sequence is 4, 9, 16, 25, ..., which is the Square numbers sequence (starting from ).
2. Pictorial Explanation:
A picture can explain why this happens. Take two consecutive triangular numbers, for example, 3 and 6.
- The number 3 can be represented by a triangle of dots with 2 dots in the base.
- The number 6 can be represented by a triangle of dots with 3 dots in the base. If you place these two triangles together (one of them rotated), they fit perfectly to form a square. The triangle of 3 dots and the triangle of 6 dots combine to form a square, which contains 9 dots. This works for any pair of consecutive triangular numbers.
Q6Section 1.4 Figure it Out
What happens when you start to add up powers of 2 starting with 1, i.e., take ? Now add 1 to each of these numbers-what numbers do you get? Why does this happen?
Solution
1. Adding up powers of 2:
Let's calculate the sums:
- The resulting sequence is 1, 3, 7, 15, ...
2. Adding 1 to each number:
- After adding 1, the new sequence is 2, 4, 8, 16, ..., which is the Powers of 2 sequence (starting from ).
3. Why this happens:
This pattern occurs because the sum of the powers of 2 up to a certain power is always one less than the next power of 2. For example:
- So, when you add 1 to a sum like , the result is simply .
Q7Section 1.4 Figure it Out
What happens when you multiply the triangular numbers by 8 and add 1? Which sequence do you get? Can you explain it with a picture?
Solution
Correction: The provided text seems to have a typo in the solution key, which refers to multiplying by 6. The question asks to multiply by 8. Let's solve the question as written.
1. The sequence:
The triangular numbers are 1, 3, 6, 10, 15, ...
Let's multiply each by 8 and add 1:
- The resulting sequence is 9, 25, 49, 81, ... This is the sequence of the squares of the odd numbers ().
2. Pictorial Explanation:
Consider a triangular arrangement of dots, for example, the number 3. If you arrange 8 such triangles around a central dot, you can form a square. For the triangular number 3, eight such arrangements plus one central dot form a square, which is 25 dots. This demonstrates the relationship pictorially.
Q8Section 1.4 Figure it Out
What happens when you start to add up hexagonal numbers, i.e., take ? Which sequence do you get? Can you explain it using a picture of a cube?
Solution
1. The sequence:
Let's calculate the sums:
- The resulting sequence is 1, 8, 27, 64, ..., which is the Cube numbers sequence ().
2. Explanation using a cube:
This pattern relates to building larger cubes from smaller ones.
- A cube is made of 1 block.
- To build a cube (8 blocks) around the initial 1 block, you need to add more blocks. This '7' is the second hexagonal number.
- To expand the cube into a cube (27 blocks), you need to add more blocks. This '19' is the third hexagonal number. So, adding the hexagonal numbers is like adding layers to a cube, with the sum being the total number of blocks in the resulting cube.
Q9Section 1.4 Figure it Out
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?
Solution
Here is one example of a pattern:
Pattern: The sum of the first 'n' counting numbers is equal to the nth triangular number.
This is the definition of triangular numbers, but we can also relate it to other sequences. For example, the sum of a counting number and the next one is the difference between their squares.
Another Pattern: Every odd number is the difference between two consecutive square numbers.
Pictorial Explanation:
Imagine a square of dots, say . If you remove a smaller square from its corner, say , the remaining shape is an L-shape (called a gnomon). The number of dots in this L-shape is . This shows that the difference between two consecutive squares is an odd number.
Q1Section 1.4 Questions from Text
By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?
Solution
Pattern: The sum of the first 'n' odd numbers is equal to .
To Find: The sum of the first 10 odd numbers.
Solution:
Here, . According to the pattern, the sum will be .
This corresponds to the total number of dots in a square grid.
Final Answer: The sum of the first 10 odd numbers is 100.
Q2Section 1.4 Questions from Text
Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?
Solution
Pattern: The sum of the first 'n' odd numbers is equal to .
To Find: The sum of the first 100 odd numbers.
Solution:
Here, . According to the pattern, the sum will be .
This corresponds to the total number of dots in a square grid.
Final Answer: The sum of the first 100 odd numbers is 10000.
Q1Section 1.5 Figure it Out
Can you recognise the pattern in each of the sequences in Table 3?
Solution
Yes, the pattern in each shape sequence can be recognized:
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Regular Polygons: The sequence starts with a regular triangle (3 sides), then a square (4 sides), a pentagon (5 sides), and so on. Each new shape is a regular polygon with one more side than the previous one.
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Complete Graphs: The sequence shows graphs where every point (vertex) is connected to every other point. The first shape has 2 vertices, the second has 3, the third has 4, and so on. Each new shape has one more vertex, with lines drawn to all other vertices.
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Stacked Squares: The sequence shows larger squares made of smaller unit squares. The first is a square, the second is a square, the third is a square, and so on. Each new shape is an grid of squares.
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Stacked Triangles: The sequence shows larger triangles made of smaller unit triangles. The first shape has a base of 1 triangle, the second has a base of 2, the third has a base of 3, and so on. Each new shape adds a new row to the base.
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Koch Snowflake: This is a fractal sequence. It starts with an equilateral triangle. To get the next shape, each straight line segment is replaced with a four-segment 'bump'. This process is repeated to create increasingly complex and detailed shapes.
Q2Section 1.5 Figure it Out
Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
Solution
Yes, the next shape in each sequence can be drawn by following its pattern.
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Regular Polygons:
- Next Shape: A regular heptagon (a polygon with 7 equal sides and angles).
- Rule: The rule is to increase the number of sides by one for each new shape, keeping all sides and angles equal.
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Complete Graphs:
- Next Shape: A shape with 5 vertices, where every vertex is connected to the other 4 vertices by a line. This will look like a pentagon with all its diagonals drawn in.
- Rule: Add one new vertex and draw a line connecting it to all of the previously existing vertices.
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Stacked Squares:
- Next Shape: A grid of small squares.
- Rule: The nth shape in the sequence is an square grid.
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Stacked Triangles:
- Next Shape: A large triangle composed of smaller triangles, with 4 small triangles forming its base.
- Rule: The nth shape is a large triangle with a base of n small triangles.
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Koch Snowflake:
- Next Shape: Take the third shape in the sequence and replace every one of its small straight line segments with the 'bump' shape. The resulting figure will be much more detailed and jagged.
- Rule: Replace every straight line segment in the current shape with a specific four-segment pattern (like a speed bump).
Q1Section 1.6 Figure it Out
Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Solution
1. Number of sides:
- Triangle: 3 sides
- Square: 4 sides
- Pentagon: 5 sides
- Hexagon: 6 sides The number sequence for the sides is 3, 4, 5, 6, ..., which is the counting numbers starting from 3.
2. Number of corners (vertices):
- Triangle: 3 corners
- Square: 4 corners
- Pentagon: 5 corners
- Hexagon: 6 corners The number sequence for the corners is also 3, 4, 5, 6, ...
3. Do you get the same sequence?
Yes, the number sequence for sides and corners is the same.
4. Why this happens:
For any simple, closed polygon, the number of sides is always equal to the number of corners (vertices). Each side connects two corners, and each corner is where two sides meet.
Q2Section 1.6 Figure it Out
Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
Solution
1. Number of lines:
The shapes shown in Table 3 for Complete Graphs likely have 2, 3, 4, and 5 vertices.
- 2 vertices: 1 line
- 3 vertices: 3 lines
- 4 vertices: 6 lines
- 5 vertices: 10 lines The number sequence for the lines is 1, 3, 6, 10, ... This is the Triangular numbers sequence.
2. Why this happens:
In a complete graph, every vertex is connected to every other vertex. For a graph with 'n' vertices, the first vertex connects to the other vertices. The second vertex needs to connect to the remaining vertices (as it is already connected to the first). This continues until the last vertex. The total number of lines is the sum , which is the formula for the th triangular number.
Q3Section 1.6 Figure it Out
How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
Solution
1. Number of little squares:
- 1st shape: square
- 2nd shape: squares
- 3rd shape: squares
- 4th shape: squares The number sequence is 1, 4, 9, 16, ... This is the Square numbers sequence.
2. Why this happens:
The nth shape in the sequence is an grid. The total number of little squares in such a grid is calculated by multiplying the number of rows by the number of columns, which is .
Q4Section 1.6 Figure it Out
How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
Solution
1. Number of little triangles:
Let's count the triangles in each row, starting from the top:
- 1st shape (base 1): 1 row with 1 triangle. Total = 1.
- 2nd shape (base 2): 2 rows with 1 and 3 triangles. Total = .
- 3rd shape (base 3): 3 rows with 1, 3, and 5 triangles. Total = .
- 4th shape (base 4): 4 rows with 1, 3, 5, and 7 triangles. Total = . The number sequence is 1, 4, 9, 16, ... This is the Square numbers sequence.
2. Why this happens:
The nth shape has 'n' rows. The number of triangles in these rows follows the sequence of the first 'n' odd numbers (). As we learned in Section 1.4, the sum of the first 'n' odd numbers is always a square number, .
Q5Section 1.6 Figure it Out
To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '-' by a 'speed bump' . As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is , i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)
Solution
1. Number of line segments:
- Shape 1 (Initial triangle): It has 3 line segments.
- Shape 2: Each of the 3 segments is replaced by 4 new segments. So, the total number of segments is .
- Shape 3: Each of the 12 segments from the previous shape is replaced by 4 new segments. So, the total is .
- Shape 4: Following the pattern, the total would be .
2. The corresponding number sequence:
The sequence is 3, 12, 48, 192, ...
The rule for this sequence is to start with 3 and multiply the previous term by 4 to get the next term. The nth term can be written as .