Class 9 Mathematics

Chapter 8 Solutions — Predicting What Comes Next: Exploring Sequences and Progressions

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Step-by-step NCERT solutions for Predicting What Comes Next: Exploring Sequences and Progressions (Chapter 8, NCERT Class 9 Mathematics) — the full working for every question, not just the final answer. You can also read the Predicting What Comes Next: Exploring Sequences and Progressions textbook chapter.

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All 35 questions in Predicting What Comes Next: Exploring Sequences and Progressions are solved in the PDF. Here's what's inside, exercise by exercise:

Sequences — Explicit and Recursive Rules

1.
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.
Find the 10th and 15th terms of the sequence tn = 5n – 3 for n ≥ 1.
Determine whether 97 and 172 are terms of the sequence tn = 5n – 3 for n ≥ 1.
Which term of the sequence tn = 5n – 3 for n ≥ 1 is 607?
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?
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.

Arithmetic Progressions

Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….
Which term of the AP: 21, 18, 15, … is –81? Also, is 0 a term of this AP? Give reasons for your answer.
Find the nth term of the AP: 11, 8, 5, 2 … Write the recursive rule for this AP.
An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term.
How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?
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?
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?

Geometric Progressions

Find the 12th term of a GP with common ratio 2, whose 8th term is 192.
Find the 10th and nth terms of the GP: 5, 25, 125, ….
(*) A sequence is given by the recursive rule t1 = 2, tn+1 = 3tn – 2 for n ≥ 1. Which term of the sequence is 730?
Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.
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. 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?
Which term of the sequence 2, 2√2, 4, … is 128?
Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 is a square sheet. To construct Stage 1, each side is trisected, points of trisection of opposite sides are joined to obtain nine smaller squares, and the centre square is removed. The same process is repeated on the eight retained shaded squares to get Stage 2, and so on. (i) How many red squares are there in Stages 0 to 3?…

Exercises

Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.
Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.
How many three-digit numbers are divisible by 7? (Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)
How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)
Find a GP for which the sum of the first two terms is -4 and the fifth term is 4 times the third term.
Find all possible ways of expressing 100 as the sum of consecutive natural numbers.
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?
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.
Find the smallest value of n such that the sum of the first n natural numbers is greater than 1,000.
Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.
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.
If the 4th, 10th and 16th terms of a GP are x, y and z respectively, prove that x, y, z are in GP.
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.
Suppose P1 = 1, P2 = 2 and for n > 2, Pn = P1 + P2 + … + P(n-1) + 1. Find the values of P1, P2, …, P8. Can you find a simpler recursive formula for Pn? Can you give an explicit formula?
Suppose W1 = 1, W2 = 2 and for n > 2, Wn = W1 + W2 + … + W(n-2) + 2. Find the values of W1, W2, …, W8. Do you recognise this sequence?

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