Class 9 Maths Ganita Manjari Chapter 8 Exercise 8.3 Solutions Predicting What Comes Next: Exploring Sequences and Progressions

Class 9 Maths Ganita Manjari Chapter 8 Exercise 8.3 Solutions provide step-by-step explanations to help students understand and solve the questions given in Exercise 8.3. This exercise is part of Chapter 8, “Predicting What Comes Next: Exploring Sequences and Progressions,” and focuses on identifying patterns, understanding sequences and applying mathematical reasoning to predict the next terms.

Students can use these Class 9 Maths Ganita Manjari Chapter 8 Exercise 8.3 Solutions to check their answers, understand the concepts involved and improve their problem-solving skills. The detailed solutions are useful for homework, revision and exam preparation, helping students develop a clear understanding of sequences and progressions.

Class 9 Maths Ganita Manjari Chapter 8 Exercise 8.3 Solutions Predicting What Comes Next: Exploring Sequences and Progressions

Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Which term of the AP : 21, 18, 15, … is – 81? Also, is 0 a term of this AP? Give reasons for your answer
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Observe the Sierpiński triangle and try to answer the following questions
(a) How many black triangles are there in Stages 0 to 3 of Fig. 8.7?
(b) Can you predict the number of black triangles at Stages 4 and 5?
(c) Can you find a rule for the number of black triangles at the nth stage? 
(d) Suppose the area of the triangle (that is, the black region) in Stage 0 is 1 square unit. What is the area of the black region in Stages 1, 2 and 3? What will be the area of the black region in Stages 4 and 5? Find a rule
for the area of the black region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10, 11 and 12? In Stage 20? At any stage?

 How is this different from the growing pattern in Fig. 8.3?

Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Harish started work at an annual salary of ₹5,00,000 and received an increment of ₹20,000 each year. After how many years did his income reach ₹7,00,000?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Find the nth term of the AP: 11, 8, 5, 2 … Write the recursive rule for this AP.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Find the 10th and nth terms of the GP: 5, 25, 125, … .
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Let us revisit the sequence tn of triangular numbers 1, 3, 6, 10, 15, … shown in Fig. 8.1. Note that the nth term of this sequence is the sum of the first n natural numbers. thus tn=n(n+1)2\rm\;t_n=\frac{n(n+1)}{2}​.
Can you use this to find the 10th, 17th and 80th triangular numbers?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Can you predict the number of squares in Stages 5 and 6 of the sequence? In Stages 10, 11 and 12? In Stage 20? At any stage?
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Let T1 = 1, T2 = 2, T3 = 4, and Tn = Tn–1 + Tn–2 + Tn–3 for n ≥ 4. Find T4, T5, T6, T7, and T8.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
A sequence is given by the recursive rule t1 = – 5, tn +1=  tn + 3 for n ≥ 1. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?
Predicting What Comes Next: Exploring Sequences and Progressions
Easy
Q.
Which term of the sequence tn = 5n – 3 for n ≥ 1 is 607?
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Determine whether 97 and 172 are terms of the sequence tn = 5n – 3 for n ≥ 1.
Predicting What Comes Next: Exploring Sequences and Progressions
Easy
Q.
Find the 10th and 15th terms of the sequence tn = 5n – 3 for n ≥ 1.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Find the first five terms of the sequence in which the nth term is given by 
(i) tn = 3n – 4, 
(ii) tn = 2 – 5n, and 
(iii) tn = n2 – 2n + 3 for n ≥ 1.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Find the 12th term of a GP with common ratio 2, whose 8th term is 192.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
A sequence is given by the recursive rule  t1= 2, tn+1 = 3tn – 2 for n ≥ 1. Which term of the sequence is 730?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the 2nd hour, 4th hour and nth hour?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Suppose W1 = 1, W2 = 2 and for n > 2,  Wn = W1 + W2 + … + Wn–2 + 2.  Find the values of W1, W2, …, W8.  Do you recognise this sequence?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Suppose P1 = 1,  P2 = 2 and for n > 2, Pn = P1 + P2 + … + Pn–1 + 1. Find the values of P1, P2, …, P8. Can you find a simpler recursive formula for Pn ? Can you give an explicit formula?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
If the 4th, 10th and 16th terms of a GP are x, y and z respectively, prove that x, y, z are in GP.
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
The sum of the first three terms of a GP is 1312\cfrac{13}{12}​  and their product is –1. Find the common ratio and the terms.
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Find the smallest value of n such that the sum of the first n natural numbers is greater than 1,000.
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Find all possible ways of expressing 100 as the sum of consecutive natural numbers.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Find a GP for which the sum of the first two terms is – 4 and the fifth term is 4 times the third term.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
How many multiples of 4 lie between 10 and 250?
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
How many three-digit numbers are divisible by 7?
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on.

Look at Fig. 8.12 and try to answer the following questions.
(i) How many red squares are there in Stages 0 to 3?
(ii) Can you the number of red squares in Stages 4 and 5?
(iii) Can you find a rule for the number of red squares at the nth stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage.
(iv) Suppose the area of the square in Stage 0 is 1 square unit.
What is the area of the red region in Stages 1, 2 and 3?
What will be the area of the red region in Stages 4 and 5?
Find the explicit as well as the recursive formula for the area of the red region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
Which term of the sequence 2, 22,\;2\sqrt2,​4, … is 128?
Predicting What Comes Next: Exploring Sequences and Progressions
Difficult
Q.
A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way — each time rising to 60% of the previous height.
(i) What height does the ball reach after the 5th bounce?
(ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the 6th time?
Predicting What Comes Next: Exploring Sequences and Progressions
Medium
Q.
Can you find the rule describing the nth term of the sequence of square numbers?

Class 9 Maths Ganita Manjari Chapter 8 Exercise 8.3 Solutions Predicting What Comes Next: Exploring Sequences and Progressions

NCERT Solutions for Class 9 Maths Chapter 8 Exercise Set 8.3

Exercise Set 8.3 focuses on geometric progressions, common ratio, nth term of a GP, bouncing ball problems and fractal patterns.

Question 1. Find the 12th term of a GP with common ratio 2, whose 8th term is 192.

Answer:
Given:

Common ratio r = 2
8th term = 192

In a GP:

t₁₂ = t₈ × r⁴

Because the 12th term is 4 places after the 8th term.

t₁₂ = 192 × 2⁴
= 192 × 16
= 3072

Final answer:
12th term = 3072

Question 2. Find the 10th and nth terms of the GP: 5, 25, 125, …

Answer:
First term:

a = 5

Common ratio:

r = 25/5 = 5

nth term of a GP:

tₙ = arⁿ⁻¹

So:

tₙ = 5 × 5ⁿ⁻¹
= 5ⁿ

10th term:

t₁₀ = 5¹⁰
= 9765625

Final answer:
tₙ = 5ⁿ and t₁₀ = 9765625

Question 3. A sequence is given by the recursive rule t₁ = 2, tₙ₊₁ = 3tₙ - 2 for n ≥ 1. Which term of the sequence is 730?

Answer:
Find the first few terms:

t₁ = 2
t₂ = 3(2) - 2 = 4
t₃ = 3(4) - 2 = 10
t₄ = 3(10) - 2 = 28
t₅ = 3(28) - 2 = 82
t₆ = 3(82) - 2 = 244
t₇ = 3(244) - 2 = 730

Final answer:
730 is the 7th term.

Question 4. Which term of the GP 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.

Answer:
First term:

a = 2

Common ratio:

r = 6/2 = 3

Explicit formula:

tₙ = 2 × 3ⁿ⁻¹

Now find n:

2 × 3ⁿ⁻¹ = 4374

3ⁿ⁻¹ = 4374/2
= 2187

2187 = 3⁷

So:

n - 1 = 7
n = 8

Recursive formula:

t₁ = 2
tₙ = 3tₙ₋₁, for n ≥ 2

Final answer:
4374 is the 8th term. Explicit formula: tₙ = 2 × 3ⁿ⁻¹. Recursive formula: t₁ = 2, tₙ = 3tₙ₋₁.

Question 5. A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell.

(i) What height does the ball reach after the 5th bounce?

Answer:
Initial height = 80 m

Each bounce reaches 60% = 0.6 of the previous height.

After 1st bounce:

80 × 0.6 = 48 m

After 5th bounce:

80 × (0.6)⁵
= 80 × 0.07776
= 6.2208 m

Final answer:
6.2208 m

(ii) What is the total vertical distance travelled by the time it hits the ground for the 6th time?

Answer:
The ball first falls 80 m.

Then it rises and falls after each bounce.

Heights reached after bounces:

1st bounce = 48 m
2nd bounce = 28.8 m
3rd bounce = 17.28 m
4th bounce = 10.368 m
5th bounce = 6.2208 m

By the time it hits the ground for the 6th time, it has:

  1. Fallen initially from 80 m.
  2. Gone up and down for the first five bounce heights.

Total distance:

= 80 + 2(48 + 28.8 + 17.28 + 10.368 + 6.2208)

= 80 + 2(110.6688)
= 80 + 221.3376
= 301.3376 m

Final answer:
301.3376 m

Question 6. Which term of the sequence 2√2, 2, 2√2/2, … is 1/128?

Answer:
The given sequence is a GP.

First term:

a = 2√2

Common ratio:

r = 1/√2

nth term:

tₙ = 2√2 × (1/√2)ⁿ⁻¹

Write in powers of 2:

2√2 = 2³ᐟ²
1/√2 = 2⁻¹ᐟ²

So:

tₙ = 2³ᐟ² × 2⁻(n-1)/2
= 2(3 - n + 1)/2
= 2(4 - n)/2
= 2²⁻ⁿᐟ²

Set:

2²⁻ⁿᐟ² = 1/128 = 2⁻⁷

So:

2 - n/2 = -7
-n/2 = -9
n = 18

Final answer:
1/128 is the 18th term.

Question 7. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet.

Diagram showing stages 0 to 3 of the Sierpiński square carpet fractal, where smaller central squares are repeatedly removed from a larger square pattern.

(i) How many red squares are there in Stages 0 to 3?

Answer:
Stage 0 has 1 red square.

At each stage, every red square is divided into 9 smaller squares and the centre square is removed. So each red square gives 8 red squares in the next stage.

Stage 0 = 1
Stage 1 = 8
Stage 2 = 8² = 64
Stage 3 = 8³ = 512

Final answer:
1, 8, 64, 512

(ii) Predict the number of red squares in Stages 4 and 5.

Answer:
Stage 4 = 8⁴ = 4096
Stage 5 = 8⁵ = 32768

Final answer:
Stage 4 = 4096, Stage 5 = 32768

(iii) Find a rule for the number of red squares at the nth stage. Write the explicit formula and recursive formula.

Answer:
At Stage n, the number of red squares is:

tₙ = 8ⁿ

This is the explicit formula if counting starts from Stage 0.

Recursive formula:

t₀ = 1
tₙ = 8tₙ₋₁, for n ≥ 1

Final answer:
Explicit formula: tₙ = 8ⁿ. Recursive formula: t₀ = 1, tₙ = 8tₙ₋₁.

(iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area in Stages 4 and 5? Find the explicit and recursive formula.

Answer:
At each stage, 8 out of 9 equal parts remain.

So, the area is multiplied by 8/9 at each stage.

Stage 0 area = 1
Stage 1 area = 8/9
Stage 2 area = (8/9)²
Stage 3 area = (8/9)³
Stage 4 area = (8/9)⁴
Stage 5 area = (8/9)⁵

Explicit formula:

sₙ = (8/9)ⁿ

Recursive formula:

s₀ = 1
sₙ = (8/9)sₙ₋₁, for n ≥ 1

As n increases, the area keeps decreasing and gets closer to 0.

Final answer:
sₙ = (8/9)ⁿ; recursive rule: s₀ = 1, sₙ = (8/9)sₙ₋₁.

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