Patterns in Mathematics Class 6 MCQs – Ganita Prakash Chapter 1

Whole numbers are 0, 1, 2, 3, … .
Number theory studies patterns and relationships among these numbers. For example, odd numbers, square numbers and powers of 2 form number sequences whose rules can be explored.

Each number is 2 more than the previous number.
Next number = 13 + 2 = 15
Adding 2 to an odd number gives the next odd number, so the sequence continues 15, 17, 19, … .

The sequence lists the even counting numbers.
First number = 2 × 1 and second number = 2 × 2.
Tenth number = 2 × 10 = 20
Counting the terms gives 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

The differences are 2, 3, 4, 5, 6 and 7.
The next difference is 8.
Next triangular number = 28 + 8 = 36
This corresponds to adding a new row of 8 dots to a triangle with 28 dots.

The numbers are 1 × 1, 2 × 2, 3 × 3, …, 7 × 7.
The next number is 8 × 8 = 64.
It can be shown as a square with 8 rows of 8 dots.

Finding patterns and their explanations is the primary focus of mathematics according to the chapter-1.

Number theory branch of mathematics studies patterns in whole numbers.

Small cubes in one layer = 5 × 5 = 25
Small cubes in 5 layers = 25 × 5 = 125
So 125 is a cube number. All three directions have 5 small cubes along them.

After the first two numbers, each number is the sum of the previous two.
8 + 13 = 21
Next number = 13 + 21 = 34
The rule uses addition of neighbouring terms rather than adding one fixed amount.

Each term is twice the previous term.
32 × 2 = 64
Next term = 64 × 2 = 128
These are powers of 2. Continuing the same rule after 128 would give 256.

Each term is obtained by multiplying the previous term by 3.
Next term = 243 × 3
= 729
The increase is not a fixed difference: the number of groups is tripled at every step.

A square arrangement is 6 × 6 = 36.
A triangular arrangement is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.
Thus, 36 belongs to both the square-number and triangular-number sequences.

The sequence 1, 3, 5, 7, … represent Odd numbers.

Square numbers sequence is formed by adding consecutive odd numbers.

Total dots = 1 + 2 + 3 + 4 + 5 + 6
= (1 + 6) + (2 + 5) + (3 + 4)
= 7 + 7 + 7 = 21
The six unequal rows form the sixth triangular number.

Dots in the smaller square = 5 × 5 = 25
Dots in the larger square = 6 × 6 = 36
New dots = 36 − 25 = 11
The new border forms an L-shaped group containing an odd number of dots.

Adding consecutive odd numbers from 1 gives square numbers.
1 + 3 + 5 + … + 19 = 10 × 10
= 100
There are 10 odd numbers in the sum, so the dots form a square with 10 rows.

Required number = 64 − 49 = 15
49 = 7 × 7 and 64 = 8 × 8.
The extra L-shaped border has 8 dots along one side and 7 more along the other, giving 8 + 7 = 15.

The odd numbers from 1 to 199 are the first 100 odd numbers.
Their sum = 100 × 100
= 10,000
This follows the same square pattern as 1 + 3 = 4 and 1 + 3 + 5 = 9.

The next number in the sequence of triangular numbers: 1, 3, 6, 10, … is 15.

The shape is associated with the sequence of regular polygons Triangle.

The numbers increase to 6 and then decrease to 1.
Sum = (1 + 2 + 3 + 4 + 5 + 6) + (1 + 2 + 3 + 4 + 5)
= 21 + 15 = 36
The result is the square number 6 × 6.

Adding counting numbers up to a number and then back down to 1 gives its square.
The highest number here is 100.
Required sum = 100 × 100 = 10,000
The middle number 100 is included only once.

The sums are 1, 2, 3, 4, … .
Each new sum contains one more 1, so its value increases by 1. These are the counting numbers, beginning with 1 and continuing without skipping a number.

The sums are 1, 3, 6, 10, … .
These are triangular numbers. Each new term adds the next counting number, just as a new row with one extra dot is added to a triangular arrangement.

15 + 21 = 36
36 = 6 × 6
The triangle with rows 1 to 5 and the triangle with rows 1 to 6 can be fitted together to make a square containing 36 dots.

The term “powers of 2” refer to in mathematics Sequences like 1, 2, 4, 8, …

A hexagon has 6 sides.

Sum of the powers of 2 = 1 + 2 + 4 + 8 + 16 + 32 = 63
Adding 1 gives 63 + 1 = 64.
The result is the next power of 2 after 32.

The increases go up by 6 each time.
Next increase = 18 + 6 = 24
Next term = 37 + 24 = 61
This represents adding the next outer group of dots to the hexagonal pattern.

Required number = 6 × 10 + 1
= 60 + 1 = 61
The same rule gives 7, 19 and 37 from the triangular numbers 1, 3 and 6. It links triangular numbers to the hexagonal pattern.

1 + 7 + 19 + 37 = 8 + 19 + 37
= 27 + 37 = 64
64 = 4 × 4 × 4
The successive sums of these hexagonal numbers are 1, 8, 27 and 64, which are cube numbers.

A regular polygon has equal-length sides and equal angles. Both conditions are needed. A regular triangle, square and regular pentagon have different numbers of sides, but each has equal sides and equal angles within that shape.

A nonagon has 9 sides.
Each pair of neighbouring sides meets at a corner, so it also has 9 corners.
In the polygon sequence, it comes after an octagon with 8 sides and before a decagon with 10 sides.

The first point can be joined to 5 others, the next to 4 new points, and so on.
Total segments = 5 + 4 + 3 + 2 + 1
= 15
This counts each connection once and gives a triangular number.

The counts are 1 × 1, 2 × 2, 3 × 3 and 4 × 4.
The fifth shape contains 5 × 5 = 25 small squares.
Each shape is a square array with one more row and one more column.

Total smallest triangles = 1 + 3 + 5 + 7
= 16
The rows contain consecutive odd numbers, whose sum is a square. With four rows, the number of smallest triangles is 4 × 4 = 16.

Each of the 48 segments gives 4 segments at the next stage.
New number of segments = 48 × 4
= 192
The segment counts follow 3, 12, 48, 192, …, with multiplication by 4 at every step.

The characteristic of square numbers is they can be visualized as dots in a square grid.

In the context of number sequences, visualization help with Understanding relationships between sequences.