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Algebra Play Exercise 6.1 Class 8 Maths – Chapter 6 NCERT Solutions

Students can refer to the NCERT Class 8 Maths Chapter 6 Algebra Play Exercise 6.1 solutions prepared by the ALLEN expert team. The solutions explain the questions clearly and can be used alongside regular practice, revision, and exam preparation.Exercise 6.1 Number pyramids and finding out the top number from the bottom row. It also contains the derivation of the formulas for 3-row and 4-row pyramids and the Virahāṅka-Fibonacci sequence by placing its numbers in number pyramids.

1.0Download Class 8 Maths Chapter 6 Ex 6.1 NCERT Solutions PDF

Download the NCERT Solutions PDF for Exercise 6.1 of Class 8 Maths Chapter 6 and use it to check your answers and review the solved questions.

Chapter 6 : Algebra Play Exercise 6.1

2.0Detailed NCERT Solutions for Algebra Play Exercise 6.1 Class 8 Maths (Ganita Prakash Term 2)

1. Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases.

4

13

8

7

11

3

10

14

25

Sol. We know that,

(1) Given, bottom row:

4

13

8

$a = 4 b = 13 c = 8$

The number in the topmost row

$= a + 2 b + c = 4 + 2 \times 13 + 8 = 4 + 26 + 8 = 38$

(2) Given, bottom row:

7

11

3

$a = 7 b = 11 c = 3$

The number in the topmost row $=$ $7 + 2 \times 11 + 3$

$= 7 + 22 + 3 = 32$

(3) Given, bottom row:

10

14

25

$a = 10, b = 14, c = 25$ The number in the topmost row $= 10 + 2 \times 14 + 25$ $= 10 + 28 + 25 = 63$

2. Write an expression for the topmost row of a pyramid with 4 rows in terms of the values in the bottom row.

Sol. Let $a, b, c, d, e, f, g, h, i$ , and j be the elements of the pyramid.

$∴ e = a + b f = b + c g = c + d h = e + f = (a + b) + (b + c) = a + 2 b + c i = f + g = (b + c) + (c + d) = b + 2 c + d j = h + i = (a + 2 b + c) + (b + 2 c + d) = a + 3 b + 3 c + d$

The resulting pyramid will be:

Thus, the expression for the top row is $(a + 3 b + 3 c + d)$ .

3. Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases.

8	19	21	13
7	18	19	6	9	7	5	11
:---	:---	:---	:---

Recall the Virahānka-Fibonacci number sequence $1, 2, 3, 5, \dots$ where each number is the sum of the two numbers before it.

Sol. (1) If $a, b, c$ and d are the bottom row, then the expression of the topmost row of the pyramid is $a + 3 b + 3 c + d$ .

Given, bottom row;

8

19

21

13

Here, $a = 8, b = 19, c = 21$ , and $d = 13$ ∴ The number in the topmost row $=$

$a + 3 b + 3 c + d = 8 + 3 (19) + 3 (21) + 13 = 8 + 57 + 63 + 13 = 141$

Thus, the number in the topmost row is 141. (2) Given, bottom row:

7

18

19

6

Here, $a = 7, b = 18, c = 19$ and $d = 6$ ∴ The number in the topmost row $=$

$a + 3 b + 3 c + d = 7 + 3 (18) + 3 (19) + 6 = 7 + 54 + 57 + 6 = 124$

Thus, the number in the topmost row is 124 . (3) Given, bottom row:

9

7

5

11

Here, $a = 9, b = 7, c = 5$ , and $d = 11$ ∴ The number in the topmost row $= a + 3 b + 3 c + d$ $= 9 + 3 (7) + 3 (5) + 11$ $= 9 + 21 + 15 + 11 = 56$ Thus, the number in the topmost row is 56 .

4. If the first three Virahānka-Fibonacci numbers are written in the bottom row of a number pyramid with three rows, fill in the rest of the pyramid. What numbers appear in the grid? What is the number at the top? Are they all Virahānka-Fibonacci numbers?

Sol. We know that the first three VirahankaFibonacci number sequence $= 1, 2, 3$ Here, the bottom row

1

2

3

Let $a, b$ , and c be the missing numbers.

b = 1 + 2 = 3

c = 2 + 3 = 5

and

a = b + c = 3 + 5 = 8

The complete pyramid is:

The numbers appear in the grid $= 1, 2, 3, 5, 8$ ∴ The number at the top $= 8$ Yes, 1, 2, 3, 5, 8 are Virahanka-Fibonacci numbers.

5. What can you say about the numbers in the pyramid and the number at the top in the following cases? (i) The first four Virahānka-Fibonacci numbers are written in the bottom row of a four-row pyramid. (ii) The first 29 Virahānka-Fibonacci numbers are written in the bottom row of a 29 -row pyramid.

Sol. (i) We know that, The first four VirahankaFibonacci numbers $= 1, 2, 3, 5$

Here, the bottom row

1

2

3

5

Let $a, b, c, d, e$ , and f be the missing numbers.

$d = 1 + 2 = 3 e = 2 + 3 = 5 f = 3 + 5 = 8 b = d + e = 3 + 5 = 8 c = e + f = 5 + 8 = 13 and a = b + c = 8 + 13 = 21$

The numbers in the pyramid are 1,2, $3, 5, 8, 13, 21, \dots$

We can say that the numbers are a Virahanka-Fibonacci sequence. $∴$ The number at the top $= 21$ Total number in bottom rowpyramid = 4

Sequence of number at the top $= 2 \times 4 - 1 = 7$ th

Fibonacci numbers $= 21$ (ii) From the above solution, we get The number at the top $= 2 \times$ (Number of Virahāṅka-Fibonacci numbers present at the bottom) - 1

$= 2 x - 1 = 2 \times 29 - 1 = (58 - 1) th = 5 7^{th} Virahanka-Fibonacci number.$

If the bottom row of an n row pyramid contains the first n Virahānka-Fibonacci numbers, what can we say about the numbers in the pyramid? What can we say about the number at the top?

Sol. When the bottom row uses the first n Virahanka-Fibonacci numbers, every number that appears anywhere in the grid will be a Virahanka-Fibonacci number. $∴$ The number at the top of the pyramid $= (2 n - 1)^{th}$ Virahanka-Fibonacci number.

3.0Key Concepts of Chapter 6 Algebra Play Exercise 6.1

Finding the top number of a 3-row number pyramid.
Expressing the top number in terms of the bottom-row values.
Finding the top number of a 4-row number pyramid.
Calculating the top number directly without filling all the intermediate rows.
Using Virahāṅka-Fibonacci numbers in number pyramids.
Identifying the sequence of numbers generated within the pyramid.
Relating the number of rows to the position of the top number in the Virahāṅka-Fibonacci sequence.

4.0Benefits of NCERT Solutions Class 8 Maths Chapter 6 Exercise 6.1

Finds the top number of a 3-row pyramid using the bottom-row values.
Forms an expression for the top number of a 4-row pyramid.
Calculates the top number directly from the bottom row.
Completes number pyramids using Virahāṅka-Fibonacci numbers.
Identifies the Virahāṅka-Fibonacci numbers appearing within the pyramid.
Connects the number of rows in the pyramid with the corresponding sequence position.
Applies the number-pyramid pattern to n rows.

NCERT Class 8 Maths Ch. 6 Algebra Play Other Exercises:-

Exercise 6.1

Exercise 6.2

Exercise 6.3

NCERT Solutions Class 8 Maths All Chapters:-

Chapter 1 : Fractions in Disguise

Chapter 2 : The Baudhayana - The Pythagoras Theorem

Chapter 3 : Proportional Reasoning - 2

Chapter 4 : Exploring Some Geometric Themes

Chapter 5 : Tales by Dots and Lines

Chapter 6 : Algebra Play

Chapter 7 : Area

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Algebra Play Exercise 6.1 Class 8 Maths – Chapter 6 NCERT Solutions

Students can refer to the NCERT Class 8 Maths Chapter 6 Algebra Play Exercise 6.1 solutions prepared by the ALLEN expert team. The solutions explain the questions clearly and can be used alongside regular practice, revision, and exam preparation.Exercise 6.1 Number pyramids and finding out the top number from the bottom row. It also contains the derivation of the formulas for 3-row and 4-row pyramids and the Virahāṅka-Fibonacci sequence by placing its numbers in number pyramids.

1.0Download Class 8 Maths Chapter 6 Ex 6.1 NCERT Solutions PDF

Download the NCERT Solutions PDF for Exercise 6.1 of Class 8 Maths Chapter 6 and use it to check your answers and review the solved questions.

Chapter 6 : Algebra Play Exercise 6.1

2.0Detailed NCERT Solutions for Algebra Play Exercise 6.1 Class 8 Maths (Ganita Prakash Term 2)

1. Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases.

4

13

8

7

11

3

10

14

25

Sol. We know that,

(1) Given, bottom row:

4

13

8

$a = 4 b = 13 c = 8$

The number in the topmost row

$= a + 2 b + c = 4 + 2 \times 13 + 8 = 4 + 26 + 8 = 38$

(2) Given, bottom row:

7

11

3

$a = 7 b = 11 c = 3$

The number in the topmost row $=$ $7 + 2 \times 11 + 3$

$= 7 + 22 + 3 = 32$

(3) Given, bottom row:

10

14

25

$a = 10, b = 14, c = 25$ The number in the topmost row $= 10 + 2 \times 14 + 25$ $= 10 + 28 + 25 = 63$

2. Write an expression for the topmost row of a pyramid with 4 rows in terms of the values in the bottom row.

Sol. Let $a, b, c, d, e, f, g, h, i$ , and j be the elements of the pyramid.

$∴ e = a + b f = b + c g = c + d h = e + f = (a + b) + (b + c) = a + 2 b + c i = f + g = (b + c) + (c + d) = b + 2 c + d j = h + i = (a + 2 b + c) + (b + 2 c + d) = a + 3 b + 3 c + d$

The resulting pyramid will be:

Thus, the expression for the top row is $(a + 3 b + 3 c + d)$ .

3. Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases.

8	19	21	13
7	18	19	6	9	7	5	11
:---	:---	:---	:---

Recall the Virahānka-Fibonacci number sequence $1, 2, 3, 5, \dots$ where each number is the sum of the two numbers before it.

Sol. (1) If $a, b, c$ and d are the bottom row, then the expression of the topmost row of the pyramid is $a + 3 b + 3 c + d$ .

Given, bottom row;

8

19

21

13

Here, $a = 8, b = 19, c = 21$ , and $d = 13$ ∴ The number in the topmost row $=$

$a + 3 b + 3 c + d = 8 + 3 (19) + 3 (21) + 13 = 8 + 57 + 63 + 13 = 141$

Thus, the number in the topmost row is 141. (2) Given, bottom row:

7

18

19

6

Here, $a = 7, b = 18, c = 19$ and $d = 6$ ∴ The number in the topmost row $=$

$a + 3 b + 3 c + d = 7 + 3 (18) + 3 (19) + 6 = 7 + 54 + 57 + 6 = 124$

Thus, the number in the topmost row is 124 . (3) Given, bottom row:

9

7

5

11

Here, $a = 9, b = 7, c = 5$ , and $d = 11$ ∴ The number in the topmost row $= a + 3 b + 3 c + d$ $= 9 + 3 (7) + 3 (5) + 11$ $= 9 + 21 + 15 + 11 = 56$ Thus, the number in the topmost row is 56 .

4. If the first three Virahānka-Fibonacci numbers are written in the bottom row of a number pyramid with three rows, fill in the rest of the pyramid. What numbers appear in the grid? What is the number at the top? Are they all Virahānka-Fibonacci numbers?

Sol. We know that the first three VirahankaFibonacci number sequence $= 1, 2, 3$ Here, the bottom row

1

2

3

Let $a, b$ , and c be the missing numbers.

b = 1 + 2 = 3

c = 2 + 3 = 5

and

a = b + c = 3 + 5 = 8

The complete pyramid is:

The numbers appear in the grid $= 1, 2, 3, 5, 8$ ∴ The number at the top $= 8$ Yes, 1, 2, 3, 5, 8 are Virahanka-Fibonacci numbers.

5. What can you say about the numbers in the pyramid and the number at the top in the following cases? (i) The first four Virahānka-Fibonacci numbers are written in the bottom row of a four-row pyramid. (ii) The first 29 Virahānka-Fibonacci numbers are written in the bottom row of a 29 -row pyramid.

Sol. (i) We know that, The first four VirahankaFibonacci numbers $= 1, 2, 3, 5$

Here, the bottom row

1

2

3

5

Let $a, b, c, d, e$ , and f be the missing numbers.

$d = 1 + 2 = 3 e = 2 + 3 = 5 f = 3 + 5 = 8 b = d + e = 3 + 5 = 8 c = e + f = 5 + 8 = 13 and a = b + c = 8 + 13 = 21$

The numbers in the pyramid are 1,2, $3, 5, 8, 13, 21, \dots$

We can say that the numbers are a Virahanka-Fibonacci sequence. $∴$ The number at the top $= 21$ Total number in bottom rowpyramid = 4

Sequence of number at the top $= 2 \times 4 - 1 = 7$ th

Fibonacci numbers $= 21$ (ii) From the above solution, we get The number at the top $= 2 \times$ (Number of Virahāṅka-Fibonacci numbers present at the bottom) - 1

$= 2 x - 1 = 2 \times 29 - 1 = (58 - 1) th = 5 7^{th} Virahanka-Fibonacci number.$

If the bottom row of an n row pyramid contains the first n Virahānka-Fibonacci numbers, what can we say about the numbers in the pyramid? What can we say about the number at the top?

Sol. When the bottom row uses the first n Virahanka-Fibonacci numbers, every number that appears anywhere in the grid will be a Virahanka-Fibonacci number. $∴$ The number at the top of the pyramid $= (2 n - 1)^{th}$ Virahanka-Fibonacci number.

3.0Key Concepts of Chapter 6 Algebra Play Exercise 6.1

Finding the top number of a 3-row number pyramid.
Expressing the top number in terms of the bottom-row values.
Finding the top number of a 4-row number pyramid.
Calculating the top number directly without filling all the intermediate rows.
Using Virahāṅka-Fibonacci numbers in number pyramids.
Identifying the sequence of numbers generated within the pyramid.
Relating the number of rows to the position of the top number in the Virahāṅka-Fibonacci sequence.

4.0Benefits of NCERT Solutions Class 8 Maths Chapter 6 Exercise 6.1

Finds the top number of a 3-row pyramid using the bottom-row values.
Forms an expression for the top number of a 4-row pyramid.
Calculates the top number directly from the bottom row.
Completes number pyramids using Virahāṅka-Fibonacci numbers.
Identifies the Virahāṅka-Fibonacci numbers appearing within the pyramid.
Connects the number of rows in the pyramid with the corresponding sequence position.
Applies the number-pyramid pattern to n rows.

Frequently Asked Questions

What is covered in Exercise 6.1 of Class 8 Maths?

How is the top number of a 3-row pyramid found?

What is the expression for a 4-row number pyramid?

Can the top number be found without completing the pyramid?

What happens when Virahāṅka-Fibonacci numbers are used in the pyramid?

How is the top number related to the Virahāṅka-Fibonacci sequence?

Where can students find NCERT Solutions for Exercise 6.1?

Join ALLEN!

Algebra Play Exercise 6.1 Class 8 Maths – Chapter 6 NCERT Solutions

1.0Download Class 8 Maths Chapter 6 Ex 6.1 NCERT Solutions PDF

2.0Detailed NCERT Solutions for Algebra Play Exercise 6.1 Class 8 Maths (Ganita Prakash Term 2)

3.0Key Concepts of Chapter 6 Algebra Play Exercise 6.1

4.0Benefits of NCERT Solutions Class 8 Maths Chapter 6 Exercise 6.1

Frequently Asked Questions

What is covered in Exercise 6.1 of Class 8 Maths?

How is the top number of a 3-row pyramid found?

What is the expression for a 4-row number pyramid?

Can the top number be found without completing the pyramid?

What happens when Virahāṅka-Fibonacci numbers are used in the pyramid?

How is the top number related to the Virahāṅka-Fibonacci sequence?

Where can students find NCERT Solutions for Exercise 6.1?

Join ALLEN!

Algebra Play Exercise 6.1 Class 8 Maths – Chapter 6 NCERT Solutions

1.0Download Class 8 Maths Chapter 6 Ex 6.1 NCERT Solutions PDF

2.0Detailed NCERT Solutions for Algebra Play Exercise 6.1 Class 8 Maths (Ganita Prakash Term 2)

3.0Key Concepts of Chapter 6 Algebra Play Exercise 6.1

4.0Benefits of NCERT Solutions Class 8 Maths Chapter 6 Exercise 6.1