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NCERT Solutions · Maths Ch 8
Predicting What Comes Next: Exploring Sequences and Progressions
35 questions · 4 exercises
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Ex 8.1
Ex 8.2
Ex 8.3
Ex 8.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
6 QUESTIONS
Exercise 8.1
Q1
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.
(3 marks)
Q2
Find the 10th and 15th terms of the sequence tₙ = 5n – 3 for n ≥ 1.
(2 marks)
Q3
Determine whether 97 and 172 are terms of the sequence tₙ = 5n – 3 for n ≥ 1.
(2 marks)
Q4
Which term of the sequence tₙ = 5n – 3 for n ≥ 1 is 607?
(1 mark)
Q5
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?
(3 marks)
Q6
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.
(2 marks)
2
7 QUESTIONS
Exercise 8.2
Q1
Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….
(2 marks)
Q2
Which term of the AP: 21, 18, 15, … is – 81? Also, is 0 a term of this AP? Give reasons for your answer.
(3 marks)
Q3
Find the nth term of the AP: 11, 8, 5, 2, … Write the recursive rule for this AP.
(2 marks)
Q4
An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term.
(3 marks)
Q5
How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?
(3 marks)
Q6
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?
(2 marks)
Q7
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?
(2 marks)
3
7 QUESTIONS
Exercise 8.3
Q1
Find the 12th term of a GP with common ratio 2, whose 8th term is 192.
(2 marks)
Q2
Find the 10th and nth terms of the GP: 5, 25, 125, … .
(2 marks)
Q3
A sequence is given by the recursive rule t₁ = 2, tₙ₊₁ = 3tₙ – 2 for n ≥ 1. Which term of the sequence is 730?
(3 marks)
Q4
Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.
(3 marks)
Q5
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?
(5 marks)
Q6
Which term of the sequence 2, 2√2, 4, … is 128?
(2 marks)
Q7
Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 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. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on. (i) How many red squares are there in Stages 0 to 3? (ii) Predict the number of red squares in Stages 4 and 5. (iii) Find a rule for the number of red squares at the nth stage (explicit and recursive). (iv) If the area of Stage 0 is 1 square unit, what is the area of the red region in Stages 1, 2 and 3? In Stages 4 and 5? Find explicit and recursive formulas for the area at the nth stage. What happens as n increases?
(5 marks)
4
EXERCISE 8.4 · 15 QUESTIONS
End-of-Chapter Exercises
Q1
Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.
(3 marks)
Q2
Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.
(3 marks)
Q3
How many three-digit numbers are divisible by 7?
(2 marks)
Q4
How many multiples of 4 lie between 10 and 250?
(2 marks)
Q5
Find a GP for which the sum of the first two terms is – 4 and the fifth term is 4 times the third term.
(3 marks)
Q6
Find all possible ways of expressing 100 as the sum of consecutive natural numbers.
(3 marks)
Q7
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?
(2 marks)
Q8
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.
(3 marks)
Q9
Find the smallest value of n such that the sum of the first n natural numbers is greater than 1,000.
(2 marks)
Q10
Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.
(3 marks)
Q11
The sum of the first three terms of a GP is 13/12 and their product is –1. Find the common ratio and the terms.
(5 marks)
Q12
If the 4th, 10th and 16th terms of a GP are x, y and z respectively, prove that x, y, z are in GP.
(2 marks)
Q13
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.
(5 marks)
Q14
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?
(3 marks)
Q15
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?
(3 marks)