Predicting What Comes Next Exploring Sequences and Progressions Class 9 MCQ Maths Chapter 8

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MCQ on Predicting What Comes Next Exploring Sequences and Progressions Class 9

Predicting What Comes Next Exploring Sequences and Progressions MCQ Class 9

Class 9 Maths MCQ Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions

Question 1.
Which of the following is the nth term formula for an arithmetic progression where the first term is 5 and the common difference is 3?
(a) T_n = 5 + 3n
(b) T_n = 5 + (n – 1) × 3
(c) T_n = 5n + 3
(d) T_n = 3n – 5
Solution:
The correct formula is T_n = 5 + (n – 1) × 3.
This is the general formula for an arithmetic progression where the first term is 5, and the common difference is 3.
Hence, (b) is the correct answer.

Question 2.
The first term of a geometric progression is 2, and the common ratio is 4. What is the 5th term of the GP?
(a) 128
(b) 64
(c) 512
(d) 16
Solution:
The nth term of a geometric progression is given by
T_n = a × r^(n-1)
Substituting a = 2, r = 4, and n = 5, we get:
T₅ = 2 × 4^5-1
= 2 × 4⁴
= 2 × 256
= 512
So, the 5th term is 512.
Hence, (c) is the correct answer.

Question 3.
The 3rd term of an arithmetic progression is 12, and the 7th term is 32. What is the common difference?
(a) 5
(b) 4
(c) 3
(d) 2
Solution:
The nth term of an arithmetic progression is given by
T_n = a + (n – 1) × d
For the 3rd term (T₃ = 12) and the 7th term (T₇ = 32), we can write:
T₃ = a + (3 – 1) × d = 12 and T₇ = a + (7 – 1) × d = 32
Solving these two equations:
a + 2d = 12 and a + 6d = 32
Subtract the first equation from the second:
(a + 6d) – (a + 2d) = 32 – 12
⇒ 4d = 20
⇒ d = 5
So, the common difference is 5.
Hence, (a) is the correct answer.

Question 4.
The nth term of a geometric progression is given by T_n = 3 × 2^n-1. What is the 4th term of this sequence?
(a) 24
(b) 48
(c) 96
(d) 192
Solution:
Substitute n = 4 into the formula
T_n = 3 × 2^n-1
⇒ T₄ = 3 × 2^4-1
⇒ T₄ = 3 × 2³
⇒ T₄ = 3 × 8
⇒ T₄ = 24
So, the 4th term is 24.
Hence, (a) is the correct answer.

Predicting What Comes Next Exploring Sequences and Progressions Class 9 Assertion and Reason Questions

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.

Question 1.
Assertion (A): In a geometric progression (GP), the ratio of consecutive terms is always constant.
Reason (R): The common ratio in a GP can be any number, positive or negative, but it is always a fixed value.
Solution:
(a) Assertion: Correct.
In a GP, the ratio of any term to its preceding term is constant, which is known as the common ratio.
Reason: Correct.
The common ratio in a GP can indeed be any real number, and it is always constant for a given sequence.
Both Assertion and Reason are correct.

Question 2.
Assertion (A): The general term of an arithmetic progression is given by T_n = a + (n – 1) × d
Reason (R): The formula for the nth term of an arithmetic progression can be used to find the common difference if the first and nth terms are known.
Solution:
(a) Assertion: Correct.
The formula T_n = a + (n – 1) × d is the general formula for the nth term of an arithmetic progression.
Reason: Correct.
If the first term a and the nth term T_n are known, the common difference d can be calculated using the rearranged formula: d = \(\frac{\mathrm{T}_n-a}{n-1}\)
Both Assertion and Reason are correct.

Question 3.
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 all have the same sign.
Solution:
(c) Assertion: Correct.
If the common ratio is negative, the terms alternate between positive and negative values.
Reason: Incorrect.
If the common ratio is negative, the terms alternate in sign, not remain the same.
Assertion is correct, Reason is incorrect.

Question 4.
Assertion (A): In an arithmetic progression, the sum of the first n terms is always positive if the first term is positive and the common difference is also positive.
Reason (R): In an arithmetic progression, the sum of the first n terms is always positive if the first term is positive and the common difference is negative.
Solution:
(c) Assertion: Correct.
If both the first term and the common difference of an arithmetic progression are positive, the terms will always increase, and the sum of the first n terms will be positive.
Reason: Incorrect.
If the common difference is negative, the terms will decrease, and the sum may become negative or even zero, depending on the values of a and n.
Assertion is correct, Reason is incorrect.

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