Predicting What Comes Next Exploring Sequences and Progressions Class 9 Worksheet with Answers Maths Chapter 8

Teachers can assign these Ganita Manjari Class 9 Worksheet and NCERT Class 9 Maths Chapter 9 Predicting What Comes Next Exploring Sequences and Progressions Worksheet with Answers Pdf for daily practice.

Predicting What Comes Next Exploring Sequences and Progressions Worksheet Class 9

Class 9 Maths Predicting What Comes Next Exploring Sequences and Progressions Worksheet

Worksheet On Predicting What Comes Next Exploring Sequences and Progressions Class 9

Multiple Choice Questions

Question 1.
If the 2^nd term of an AP is 13 and the 5^th term is 25, what is its 7^th term?
(a) 30
(b) 33
(c) 37
(d) 38
Answer:
(b) 33

Question 2.
If 7 times the 7^th term of an AP is equal to 11 times its 11^th term, then its 18th term will be
(a) 11
(b) 18
(c) O
(d) 7
Answer:
(c) O

Question 3.
The sum of first 8 multiples of 8 is _____ .
(a) 264
(b) 188
(c) 252
(d) 288
Answer:
(d) 288

Question 4.
What is the common difference of an AP in which a₂₀ – a₁₄ = 36?
(a) 5
(b) 6
(c) 8
(d) 9
Answer:
(b) 6

Question 5.
In a GP, each term is obtained by
(a) adding a fixed number
(b) multiplying by a fixed number
(c) subtracting a fixed number
(d) none of these
Answer:
(b) multiplying by a fixed number

Question 6.
Which of the following forms an AP?
(a) 100,90,80,70
(b) \(\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}\)
(c) Both (a) and (b)
(d) None of these
Answer:
(c) Both (a) and (b)

Question 7.
The n^th term of a sequence is given by t_n = 3n². The ratio of the ^rd term to the 2^nd term is ___________.
(a) 2 : 1
(b) 3 : 2
(c) 9 : 4
(d) 1 : 3
Answer:
(c) 9 : 4

Question 8.
A recursive rule generates a sequence where each term increases by 5. If t_n = 20, the first term is _______.
(a) 0
(b) 5
(c) 10
(d) 15
Answer:
(b) 5

Question 9.
A recursive sequence is defined as t₁ = 2, t_n = 3t_n-1 ; n ≥ 2. Which statement is correct?
(a) It is an AP with common ratio 3
(b) It is a GP with common ratio 3
(c) It is neither AP nor GP
(d) It has a common difference
Answer:
(b) It is a GP with common ratio 3

Question 10.
The n^th term of a geometric progression is given by t_n = 2 × (-3)^n-1 ; n ≥ 2. What is the 5^th term of the sequence?
(a) -54
(b) 54
(c) -162
(d) 162
Answer:
(d) 162

Question 11.
The 7^th term of the GP 3, 6, 12, …, is ______.
(a) 48
(b) 192
(c) 96
(d) 144
Answer:
(b) 192

Question 12.
A GP has 3^rd term =12 and 5^th term = 48. The common ratio is ________.
(a) 2
(b) 3
(c) 4
(d) 6
Answer:
(a) 2

Question 13.
Which statement is true about the Virahanka-Fibonacci sequence?
(a) It has a constant difference
(b) It has a constant ratio
(c) Each term depends on previous two terms
(d) It is a decreasing sequence
Answer:
(c) Each term depends on previous two terms

Question 14.
In a Sierpinski Triangle, the number of black triangles increases but the total shaded area with each stage. This happens because
(a) the number follows an AP and the area follows a GP
(b) the number is multiplied by 3 and the area is multiplied by \(\frac{3}{4}\)
(c) both number and area are multiplied by 3
(d) both number and area remain constant
Answer:
(b) the number is multiplied by 3 and the area is multiplied by \(\frac{3}{4}\)

Question 15.
A recursive sequence is defined as t_n = 5, t_n = f_n-1 – 2, n ≥ 2. Find the value of n for which t_n = -5
(a) 4
(b) 5
(c) 6
(d) 7
Answer:
(c) 6

Assertion-Reason

In Q. 1 to 5, two statements are given, one labelled as Assertion (A) and the other labelled as Reason (R). Select the correct answer from the options (a), (b), (c), and (d) given below.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

1. Assertion (A): Recursive rules depend on previous terms.
Reason (R): They require earlier values to find the next term.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

2. Assertion (A): In a geometric progression, if the common ratio is negative, the sequence alternates between positive and negative terms.
Reason (R): A geometric progression with a negative common ratio will result in terms that have the same sign.
Answer:
(c) Assertion (A) is true, but Reason (R) is false.

3. Assertion (A): The difference between any two consecutive terms in the sequence of numbers √6, √24, √54, √96,… is 3√6.
Reason (R): The sequence of numbers √6, √24, √54, √96,… form an arithmetic progression.
Answer:
(d) Assertion (A) is false, but Reason (R) is true.

4. Assertion (A): The 4th term of the sequence defined by t_n = 2t_n-1 + 3 with t₁ = 1 is 29.
Reason (R): Each term depends on doubling the previous term and adding 3.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

5. Assertion (A): If t_n = 5 × \(\left(\frac{1}{2}\right)^{n-1}\), the sequence is decreasing.
Reason (R): The common ratio is less than 1.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

Predicting What Comes Next Exploring Sequences and Progressions Class 9 Ganita Manjari Worksheet

Short Answer Type Questions – I

Question 1.
If the 4^th term of an AP is 18 and common difference is 4, find the first term.
Answer:
6

Question 2.
Which term of the AP: 4, 9, 14, … is 89?
Answer:
18^th term

Question 3.
Find the value of x, so that x + 2, 4x – 6 and 3x – 2 are the three consecutive terms of an arithmetic progression.
Answer:
3

Question 4.
Consider the sequence defined by the explicit formula t_n = 3n – 2, where n is a natural number. Write the first 5 terms of the sequence.
Answer:
[1, 4, 7, 10, 13]

Question 5.
Find the n^th term of the geometric progression in which the second term is 12 and the fourth term is 108.
Answer:
4(3)_n-1 or -4(-3)_n-1

Question 6.
Write the recursive formula of a sequence where each term is 3 more than twice the previous term.
Answer:
t_n = 2t_n-1 + 3; n ≥ 2

Question 7.
If the n^th term of a GP is 128 and both the first term a and the common ratio r are 2. Find the value of n.
Answer:
7

Short Answer Type Questions – II

Question 1.
Mihir started working in 2020 at an annual salary of ₹ 600000 and received an increment of ₹ 50000 each year. In which year will his income reach ₹ 1000000?
Answer:
2028

Question 2.
A library assigns codes to books using 2-digit numbers divisible by 5. Find how many such codes exist and find their total sum.
Answer:
18,945

Question 3.
An AP consists of 40 terms in which the 3^rd term is 14 and the last term is 125. Find the 29^th term.
Answer:
92

Question 4.
Find the recursive rule for the following arithmetic progressions.
(a) 12, 7, 2, -3 ….
(b) [/latex]\frac{3}{2}, \frac{7}{2}, \frac{11}{2}, \frac{15}{2}[/latex]
Answer:
(a) t₁ = 12, t_n = t_n-1 – 5, n ≥ 2
(b) t₁ = \(\frac{3}{2}\), t_n = t_n-1 + 2 n ≥ 2

Question 5.
Which term of the progression 18, -12, 8, … is \(\frac{512}{729}\)?
Answer:
9^th term

Question 6.
If the 4^th, 8^th, and 12^th terms of a GP are a, b, and c respectively, then prove that b = √ac.

Question 7.
A computer virus starts by infecting 5 files. In every subsequent hour, the number of infected files doubles, forming the sequence 5, 10, 20, … At which hour will 640 files be infected? Write the explicit and recursive formulas for this growth.
Answer:
8^th hour; Recursive formula: t_n = 5(2)^n-1;
Recursive formula: t₁ = 5, t_n= 2t_n-1, n ≥ 2

Long Answer Type Questions

Question 1.
The sum of the first three terms of a geometric progression is \(\frac{39}{10}\), and their product is 1. Find the common ratio and terms.
Answer:
For common ratio \(\frac{5}{2}\), terms are \(\frac{2}{5}\), 1, \(\frac{5}{2}\)
For common ratio \(\frac{2}{5}\), terms are \(\frac{5}{2}\), 1, \(\frac{2}{5}\)

Question 2.
If 10 times the 10^th term of an AP is equal to 15 times the 15^thterm, show that 25th term of the AP is zero.
Answer:
0

Question 3.
The sum of the 5^th and 9^th terms of an AP is 40, and the sum of the 7^th and 11^th terms is 60. Find the sum of first five terms of the AP.
Answer:
0

Question 4.
Evaluate:
(a) The sum of all integers from 1 to 500 that are multiples of both 3 and 4.
(b) The sum of all integers from 1 to 500 that are multiples of either 3 or 4.
Answer:
(a) 10332
(b) 62751

Question 5.
10 trees are planted in a straight line at equal intervals of 5 metres. To water them, the gardener must bring water for each tree separately from a well 10 metres from the first tree in line with the trees. How much distance will he have to cover in order to water all the trees, if he starts from the well?
Answer:
650 metres

Case-Based Questions

Question 1.
Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of cable , car ride from base point to topmost point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which the car moves. The distance of first pole from the base point is 200 m and subsequent poles are installed at equal interval of 150 m. Further, the distance of last pole from the top is 300 m.
Based on above information, answer the following questions using Arithmetic Progression:
(a) Find the distance of the 10^th pole from the base.
Answer:
1550 metres

(b) Find the distance between the 15^th pole and 25^th pole.
Answer:
1500 metres

(c) Find the time taken by cable car to reach the 15^th pole from the top if it is moving at the speed of 5m/sec.
Answer:
9 minutes
OR
Find the total number of poles installed along the entire journey.
Answer:
31 poles

Question 2.
Aryan is studying the Sierpinski triangle, a well-known fractal pattern. The pattern begins with an equilateral triangle (Stage 0), and at each stage, the triangle is divided into 4 smaller triangles. The central triangle is removed, leaving 3 smaller triangles. Aryan has observed the progression of the Sierpinski triangle from Stage 0 to Stage 3 and is now trying to understand the number of black triangles and the area of the black region at each stage.

(a) How many black triangles are there from Stage O to Stage 3 of the Sierpinski triangle?
Answer:
[1, 3, 9, 27]

(b) At which stage will the number of black triangles be 81?
Answer:
Stage 4

(c) Write a rule for the number of black triangles at the n^th stage.
Answer:
3ⁿ
OR
If the area of the black region at Stage 0 is 1 square unit, what is the area of the black region at Stage 5?
Answer:
\(\frac{243}{1024}\) sq.unit

Competency Based Questions

Question 1.
Harsha made a wind chime using a frame and metal rods. She punched 8 holes in the frame, each 2 cm apart, and then hung 6 metal rods from the frame, as shown in the figure. The ends of the metal rods are aligned over a line, shown by the dotted line in the figure. If all of the rods are straight and not swaying, then what is the length of Rod P?

Answer:
\(\frac{111}{7}\) cm

Question 2.
The 3^rd and the 14^th terms of an arithmetic progression are (-9) and (35) respectively. Which term of this arithmetic progression is five times the 6^th term?
Answer:
9^th term

Question 3.
A sequence is defined by the recursive rule: t₁ = 2, t_n = 2 x t_n-1 for n ≥ 2. Find the 10^th term of the sequence.
Answer:
1024

Question 4.
A sequence is defined recursively by the rule t₁ = 5 and t_n = 2t_n-1 + 3 for n ≥ 2. Calculate the 7^th term in the sequence. Also, prove that the general formula for the n^th term of the sequence is tⁿ = 8 × 2_n-1 -3.
Answer:
t₇ = 509

Question 5.
Rakesh was given his pocket money on Jan 1st, 2024. From this money, he puts ₹1 on Day 1, ₹2 on Day 2, ₹3 on Day 3, and continued doing so till the end of the month into his piggy bank. He also spent ₹204 of his pocket money and found that at the end of the month he still had ₹100 with him. How much was his pocket money for the month?
Answer:
₹ 800

The post Predicting What Comes Next Exploring Sequences and Progressions Class 9 Worksheet with Answers Maths Chapter 8 appeared first on Learn CBSE.

from Learn CBSE https://ift.tt/3qDkA9a
via IFTTT