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Predicting What Comes Next: Exploring Sequences and Progressions
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NCERT Solutions
Predicting What Comes Next: Exploring Sequences and Progressions
44 Solutions
Exercise:
All Exercises
End-of-Chapter Exercises
Exercise Set 8.1
Exercise Set 8.2
Exercise Set 8.3
Exercises
Q1
End-of-Chapter Exercises
Find the
31
st
31^{\text {st }}
3
1
st
term of an AP whose
11
th
11^{\text {th }}
1
1
th
term is 38 and
16
th
16^{\text {th }}
1
6
th
term is 73.
Q2
End-of-Chapter Exercises
Determine the AP whose third term is 16 and whose
7
th
7^{\text {th }}
7
th
term exceeds the
5
th
5^{\text {th }}
5
th
term by 12.
Q3
End-of-Chapter Exercises
*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.)
Q4
End-of-Chapter Exercises
*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.)
Q5
End-of-Chapter Exercises
*Find a GP for which the sum of the first two terms is -4 and the fifth term is 4 times the third term.
Q6
End-of-Chapter Exercises
*Find all possible ways of expressing 100 as the sum of consecutive natural numbers.
Q7
End-of-Chapter Exercises
*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
2
nd
2^{\text {nd }}
2
nd
hour,
4
th
4^{\text {th }}
4
th
hour and
n
th
n^{\text {th }}
n
th
hour?
Q8
End-of-Chapter Exercises
*The sum of the
4
th
4^{\text {th }}
4
th
and
8
th
8^{\text {th }}
8
th
terms of an AP is 24 and the sum of the
6
th
6^{\text {th }}
6
th
and
10
th
10^{\text {th }}
1
0
th
terms is 44 . Find the first three terms of the AP.
Q9
End-of-Chapter Exercises
*Find the smallest value of
n
n
n
such that the sum of the first
n
n
n
natural numbers is greater than 1,000.
Q10
End-of-Chapter Exercises
*Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit formula as well as the recursive formula for the
n
th
n^{\text {th }}
n
th
term.
Q11
End-of-Chapter Exercises
*The sum of the first three terms of a GP is
13
12
\frac{13}{12}
12
13
and their product is -1. Find the common ratio and the terms.
Q12
End-of-Chapter Exercises
*If the
4
th
,
10
th
4^{\text {th }}, 10^{\text {th }}
4
th
,
1
0
th
and
16
th
16^{\text {th }}
1
6
th
terms of a GP are
x
,
y
x, y
x
,
y
and
z
z
z
respectively, prove that
x
,
y
,
z
x, y, z
x
,
y
,
z
are in GP.
Q13
End-of-Chapter Exercises
*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.
Q14
End-of-Chapter Exercises
*Suppose
P
1
=
1
,
P
2
=
2
\mathrm{P}_{1}=1, \mathrm{P}_{2}=2
P
1
=
1
,
P
2
=
2
and for
n
>
2
,
P
n
=
P
1
+
P
2
+
⋯
+
P
n
−
1
+
1
n>2, \mathrm{P}_{\mathrm{n}}=\mathrm{P}_{1}+\mathrm{P}_{2}+\cdots+\mathrm{P}_{\mathrm{n}-1}+1
n
>
2
,
P
n
=
P
1
+
P
2
+
⋯
+
P
n
−
1
+
1
. Find the values of
P
1
,
P
2
,
…
,
P
8
\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{8}
P
1
,
P
2
,
…
,
P
8
. Can you find a simpler recursive formula for
P
n
\mathrm{P}_{\mathrm{n}}
P
n
? Can you give an explicit formula?
Q15
End-of-Chapter Exercises
*Suppose
W
1
=
1
,
W
2
=
2
\mathrm{W}_{1}=1, \mathrm{W}_{2}=2
W
1
=
1
,
W
2
=
2
and for
n
>
2
,
W
n
=
W
1
+
W
2
+
⋯
+
W
n
−
2
+
2
n>2, \mathrm{W}_{\mathrm{n}}=\mathrm{W}_{1}+\mathrm{W}_{2}+\cdots+ \mathrm{W}_{\mathrm{n}-2}+2
n
>
2
,
W
n
=
W
1
+
W
2
+
⋯
+
W
n
−
2
+
2
. Find the values of
W
1
,
W
2
,
…
,
W
8
\mathrm{W}_{1}, \mathrm{W}_{2}, \ldots, \mathrm{W}_{8}
W
1
,
W
2
,
…
,
W
8
. Do you recognise this sequence?
Q1
Exercise Set 8.1
Find the first five terms of the sequence in which the
n
th
n^{\text {th }}
n
th
term is given by (i)
t
n
=
3
n
−
4
t_{n}=3 n-4
t
n
=
3
n
−
4
, (ii)
t
n
=
2
−
5
n
t_{n}=2-5 n
t
n
=
2
−
5
n
, and (iii)
t
n
=
n
2
−
2
n
+
3
t_{n}=n^{2}-2 n+3
t
n
=
n
2
−
2
n
+
3
for
n
≥
1
n \geq 1
n
≥
1
.
Q2
Exercise Set 8.1
Find the
10
th
10^{\text {th }}
1
0
th
and
15
th
15^{\text {th }}
1
5
th
terms of the sequence
t
n
=
5
n
−
3
t_{n}=5 n-3
t
n
=
5
n
−
3
for
n
≥
1
n \geq 1
n
≥
1
.
Q3
Exercise Set 8.1
Determine whether 97 and 172 are terms of the sequence
t
n
=
5
n
−
3
t_{n}=5 n-3
t
n
=
5
n
−
3
for
n
≥
1
n \geq 1
n
≥
1
.
Q4
Exercise Set 8.1
Which term of the sequence
t
n
=
5
n
−
3
t_{n}=5 n-3
t
n
=
5
n
−
3
for
n
≥
1
n \geq 1
n
≥
1
is 607?
Q5
Exercise Set 8.1
A sequence is given by the recursive rule
t
1
=
−
5
,
t
n
+
1
=
t
n
+
3
t_{1}=-5, t_{n+1}=t_{n}+3
t
1
=
−
5
,
t
n
+
1
=
t
n
+
3
for
n
≥
1
n \geq 1
n
≥
1
. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?
Q6
Exercise Set 8.1
Let
T
1
=
1
,
T
2
=
2
,
T
3
=
4
\mathrm{T}_{1}=1, \mathrm{~T}_{2}=2, \mathrm{~T}_{3}=4
T
1
=
1
,
T
2
=
2
,
T
3
=
4
, and
T
n
=
T
n
−
1
+
T
n
−
2
+
T
n
−
3
\mathrm{T}_{\mathrm{n}}=\mathrm{T}_{\mathrm{n}-1}+\mathrm{T}_{\mathrm{n}-2}+\mathrm{T}_{\mathrm{n}-3}
T
n
=
T
n
−
1
+
T
n
−
2
+
T
n
−
3
for
n
≥
4
n \geq 4
n
≥
4
. Find
T
4
,
T
5
,
T
6
,
T
7
\mathrm{T}_{4}, \mathrm{~T}_{5}, \mathrm{~T}_{6}, \mathrm{~T}_{7}
T
4
,
T
5
,
T
6
,
T
7
, and
T
8
\mathrm{T}_{8}
T
8
.
Q1
Exercise Set 8.2
Find the
10
th
10^{\text {th }}
1
0
th
and
26
th
26^{\text {th }}
2
6
th
terms of the AP: 3, 8, 13, 18, ....
Q2
Exercise Set 8.2
Which term of the AP : 21, 18, 15, ... is - 81? Also, is 0 a term of this AP? Give reasons for your answer.
Q3
Exercise Set 8.2
Find the
n
th
n^{\text {th }}
n
th
term of the AP: 11, 8, 5, 2 ... Write the recursive rule for this AP.
Q4
Exercise Set 8.2
An AP consists of 50 terms in which the
3
rd
3^{\text {rd }}
3
rd
term is 12 and the last term is 106 . Find the
29
th
29^{\text {th }}
2
9
th
term. (Hint: If '
a
a
a
' is the first term and ' d ' the common difference, then we arrive at the equations
a
+
2
d
=
12
a+2 d=12
a
+
2
d
=
12
and
a
+
49
d
=
106
a+49 d=106
a
+
49
d
=
106
. Solve this pair of linear equations for '
a
a
a
' and '
d
d
d
'.)
Q5
Exercise Set 8.2
How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?
Q6
Exercise Set 8.2
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?
Q7
Exercise Set 8.2
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?
Q1
Exercise Set 8.3
Find the
12
th
12^{\text {th }}
1
2
th
term of a GP with common ratio 2, whose
8
th
8^{\text {th }}
8
th
term is 192.
Q2
Exercise Set 8.3
Find the
10
th
10^{\text {th }}
1
0
th
and
n
th
n^{\text {th }}
n
th
terms of the GP: 5, 25, 125, ... .
Q3
Exercise Set 8.3
*A sequence is given by the recursive rule
t
1
=
2
,
t
n
+
1
=
3
t
n
−
2
t_{1}=2, t_{n+1}=3 t_{n}-2
t
1
=
2
,
t
n
+
1
=
3
t
n
−
2
for
n
≥
1
n \geq 1
n
≥
1
. Which term of the sequence is 730?
Q4
Exercise Set 8.3
Which term of the GP: 2, 6, 18, ... is 4374? Write the explicit formula as well as the recursive formula for the
n
th
n^{\text {th }}
n
th
term.
Q5
Exercise Set 8.3
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
5
th
5^{\text {th }}
5
th
bounce?
(ii)
What is the total vertical distance the ball has travelled by the time it hits the ground for the
6
e
x
t
t
h
6{ }^{ ext {th }}
6
e
x
t
t
h
time?
Q6
Exercise Set 8.3
Which term of the sequence
2
,
2
2
,
4
,
…
2,2 \sqrt{2}, 4, \ldots
2
,
2
2
,
4
,
…
is 128 ?
Q7
Exercise Set 8.3
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 predict the number of red squares in Stages 4 and 5?
(iii)
Can you find a rule for the number of red squares at the
n
th
n^{\text {th }}
n
th
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
n
th
n^{\text {th }}
n
th
stage. What happens to this area as
n
n
n
, the number of stages, goes on increasing?
Q1
Exercises
Consider the sequence 1, 4, 7, 10, 13, ... Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is
1
+
4
+
7
=
12
1+4+7=12
1
+
4
+
7
=
12
, and so on.)
Q2
Exercises
Can you write
t
5
,
t
6
,
t
7
t_{5}, t_{6}, t_{7}
t
5
,
t
6
,
t
7
and
t
8
\mathrm{t}_{8}
t
8
for the sequence of triangular numbers?
Q3
Exercises
Using the explicit rule
u
n
=
2
n
−
1
u_{n}=2 n-1
u
n
=
2
n
−
1
, find the
53
rd
53^{\text {rd }}
5
3
rd
term, the
108
th
108^{\text {th }}
10
8
th
term, and the
1170
th
1170{ }^{\text {th }}
1170
th
term of the odd number sequence.
Q4
Exercises
Consider the expression
t
n
=
3
n
−
7
t_{n}=3 n-7
t
n
=
3
n
−
7
.
(i)
Find its first, second, third,
12
th
,
18
th
12^{\text {th }}, 18^{\text {th }}
1
2
th
,
1
8
th
and
50
th
50^{\text {th }}
5
0
th
terms. *
(ii)
Which term of the sequence is 332? *
(iii)
Is 557 a term of this sequence? Why or why not?
Q5
Exercises
Verify that the following sequences are arithmetic progressions and write their
n
th
n^{\text {th }}
n
th
terms. What do you observe when you plot the ordered pairs emerging from them?
(i)
2, 5, 8, 11, ...
(ii)
-5, -1, 3, 7, ...
Q6
Exercises
Using the formula
t
n
=
a
+
(
n
−
1
)
×
d
t_{n}=a+(n-1) \times d
t
n
=
a
+
(
n
−
1
)
×
d
, find the
n
th
n^{\text {th }}
n
th
term of the following arithmetic progressions.
(i)
1
2
,
5
2
,
9
2
,
13
2
,
…
\frac{1}{2}, \frac{5}{2}, \frac{9}{2}, \frac{13}{2}, \ldots
2
1
,
2
5
,
2
9
,
2
13
,
…
(ii)
1.5, 3.5, 5.5, 7.5, ...
Q7
Exercises
Find recursive rules for the APs in the previous exercises.
Q8
Exercises
Check whether the following sequences are geometric progressions and find their
n
th
n^{\text {th }}
n
th
terms.
(i)
2
,
10
,
50
,
250
,
…
2,10,50,250, \ldots
2
,
10
,
50
,
250
,
…
(ii)
4
,
8
3
,
16
9
,
32
27
,
…
4, \frac{8}{3}, \frac{16}{9}, \frac{32}{27}, \ldots
4
,
3
8
,
9
16
,
27
32
,
…
(iii)
3
,
−
3
2
,
3
4
,
−
3
8
,
…
3, \frac{-3}{2}, \frac{3}{4}, \frac{-3}{8}, \ldots
3
,
2
−
3
,
4
3
,
8
−
3
,
…
Q9
Exercises
Can you find a recursive rule for the formula
t
n
=
3
×
10
n
−
1
t_{n}=3 \times 10^{n-1}
t
n
=
3
×
1
0
n
−
1
that generates the geometric progression
3
,
30
,
300
,
3000
,
…
3,30,300,3000, \ldots
3
,
30
,
300
,
3000
,
…
?
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