patterns in mathematics solution

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class 6 math chapter 1 patterns in mathematics solution you can get here for detail solution can download in pdf and you can watch video also

1.PATTERNS IN MATHEMATICS Solution

1.Can you recognise the pattern in each of the sequences in Table 1?  

Ans:

(a) 1, 1, 1, 1, 1, 1, 1, ……

The number ‘1’ is repeated indefinitely.

(b) 1, 2, 3, 4, 5, 6, 7, ……

Counting numbers starting from ‘1’. Each number increases by 1.

(c) 1, 3, 5, 7, 9, 11, 13, ……

Odd numbers. Start from 1 and keep adding 2.

(d) 2, 4, 6, 8, 10, 12, 14, ……

Even numbers. Start from 2 and keep adding 2.

(e) 1, 3, 6, 10, 15, 21, 28, ……

Triangular numbers. Each term is the sum of the natural numbers up to that term.

(f) 1, 4, 9, 16, 25, 36, 49, ……

Square numbers. Each term is the square of its position (12, 22, 32, …).

(g) 1, 8, 27, 64, 125, 216, ……

Cube numbers. Each term is the cube of its position ((12, 22, 32, …).

(h) 1, 2, 3, 5, 8, 13, 21, ……

Virahānka numbers (Fibonacci sequence). Each number is the sum of the two preceding numbers

(i) 1, 2, 4, 8, 16, 32, 64, ……

Powers of 2. Each term is raised to the power of its position (20, 21, 22, …).

(j) 1, 3, 9, 27, 81, 243, 729, ……

Powers of 3. Each term is 3 raised to the power of its position (30, 31, 32, …).

2. 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 forformingthenumbers in the sequence.  

Ans:

(a) 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …

The sequence consists of repeating the number 1. The next three numbers are also 1.

(b) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

This is a sequence of consecutive counting numbers. The next three numbers are 8, 9, and 10.

(c) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …

These are consecutive odd numbers, increasing by 2. The next three numbers are 15, 17, and 19.

(d) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …

The sequence contains even numbers, increasing by 2. The next three numbers are 16, 18, and 20.

(e) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

A sequence of triangular numbers where each number is the sum of the natural numbers up to a certain point.
28 + 8 = 36
36 + 9 = 45
45 + 10 = 55

(f) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

Square numbers are formed by multiplying a number by itself. The next three are 64, 81, and 100.
82 = 64
92 = 81
102 = 100

(g) 1, 8, 27, 64, 125, 216, 343, 512, 729, …

A sequence of cube numbers where each number is the cube of a natural number.

The next three numbers are 343, 512, and 729.
73 = 343
83 = 512
93 = 729

(h) 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

This is the Fibonacci (Virahanka) sequence, where each number is the sum of the previous two. The next three numbers are 34, 55, and 89.
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89

(i) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, …

This shows powers of 2, where each number is multiplied by 2. The next three numbers are 128, 256, and 512.
64 × 2 = 128
128 × 2 = 256
256 × 2 = 512

(j) 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, …

This shows powers of 3, where each number is multiplied by 3. The next three numbers are 2187, 6561, and 19683.
729 × 3 = 2187
2187 × 3 = 6561
6561 × 3 = 19683

Question 2.
(a) Why are 1, 3, 6, 10, 15,…… called triangular numbers?
(b) Why are 1, 4, 9, 16, 25,…….. called square numbers or squares?
(c) Why are 1, 8, 27, 64, 125,…….. called cubes?
Solution:
(a) Triangular numbers are called so because these numbers of dots can be arranged to make a triangle.
1, 3, 6, 10,…….. are triangular numbers.
1, 3, 6, 10,……… dots can be arranged to make a triangle

3. 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:
Three other such numbers are 1, 1225, and 41616.
1225 is the 49th triangular number.
Also, 1225 can be represented by a square having 36 dots along its side.

4. What would you call the following sequence of numbers?

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence

1st number = 1
2nd number = 1 + 6 = 7 (2nd number = 1st number + 6 × 1)
3rd number = 7 + 12 = 19 (3rd number = 2nd number + 6 × 2)
4th number = 19 + 18 = 37 (4th number = 3rd number + 6 × 3)
5th number = 37 + 24 = 61 (5th number = 4th number + 6 × 4)
Hence, the next number in the sequence is 61.

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