08 - Question bank - Arithmetic Progression - Class 10 - Maths Class 10 Notes | EduRev
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Class 10 : 08 - Question bank - Arithmetic Progression - Class 10 - Maths Class 10 Notes | EduRev
Page 1
1) Find 6
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 6
6
6 1 ? ? ?
?
a
216 ) 1 (
5
6
? ? ? a = -216
2) Find 13
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 13
13
13 1 ? ? ?
?
a
2197 ) 1 (
12
13
? ? ? a = 2197
3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10.
Find the value of the 20
th
term.
Solution:
Here, common difference (d) = -10
First term 200
1
? a
d n a a
n
) 1 (
1
? ? ?
) 10 )( 1 20 ( 200
20
? ? ? ? a
10
20
? a
4) An arithmetic sequence has a common difference equal to 10 and its 6
th
term is equal to 52. Find its
15
th
term
Solution:
We use nth term formula d n a a
n
) 1 (
1
? ? ? for the 6
th
term
) 1 6 ( 10
52
1 6
6
? ? ?
?
a a
a
50 52
1
? ?a
Therefore, 2 50 52
1
? ? ? a
Now, we know the first term and the common difference, we use the nth term formula to find the
15
th
term.
142 ) 1 15 ( 10 2
15
? ? ? ? a
Arithmetic Progressions
Page 2
1) Find 6
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 6
6
6 1 ? ? ?
?
a
216 ) 1 (
5
6
? ? ? a = -216
2) Find 13
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 13
13
13 1 ? ? ?
?
a
2197 ) 1 (
12
13
? ? ? a = 2197
3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10.
Find the value of the 20
th
term.
Solution:
Here, common difference (d) = -10
First term 200
1
? a
d n a a
n
) 1 (
1
? ? ?
) 10 )( 1 20 ( 200
20
? ? ? ? a
10
20
? a
4) An arithmetic sequence has a common difference equal to 10 and its 6
th
term is equal to 52. Find its
15
th
term
Solution:
We use nth term formula d n a a
n
) 1 (
1
? ? ? for the 6
th
term
) 1 6 ( 10
52
1 6
6
? ? ?
?
a a
a
50 52
1
? ?a
Therefore, 2 50 52
1
? ? ? a
Now, we know the first term and the common difference, we use the nth term formula to find the
15
th
term.
142 ) 1 15 ( 10 2
15
? ? ? ? a
Arithmetic Progressions
5) If x times the x
th
term of an AP is equal to y times its y
th
term , Prove that the (x + y)
th
term of the
AP is zero
Solution:
Let ‘a’ be the first term and‘d’ be the common difference of an AP
By the given condition,
y x
yt xt ?
? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ?
? ?
? ?
? ?
0
0 ........ 0 ) 1 (
0 ) 1 ( ) (
0 ) ( ) )( ( ) (
0 ) ( ) (
0 ) ( ) (
) ( ) (
2 2
2 2
2 2
? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
?y x
t
x d y x a
d y x a y x
d y x y x y x a y x
d y x y x a y x
d y y d x x ya xa
d y y ya d x x xa
?
6) Which term of AP 7, 14, 21,……..is 133
Solution:
Here, a = 7 , d = 14 – 7= 7
Let 133 be the
th
n term of the AP ( Where N n ? )
133 ?
n
t
133 ) 1 ( ? ? ? ? d n a
? ? N n
n
n
n
n
? ? ?
? ?
? ?
? ? ? ?
? ? ? ?
19 . .......... 19
7
133
133 7
133 7 7 7
133 7 ) 1 ( 7
Therefore, 133 is the 19
th
term of the given AP.
Arithmetic Progressions
Page 3
1) Find 6
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 6
6
6 1 ? ? ?
?
a
216 ) 1 (
5
6
? ? ? a = -216
2) Find 13
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 13
13
13 1 ? ? ?
?
a
2197 ) 1 (
12
13
? ? ? a = 2197
3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10.
Find the value of the 20
th
term.
Solution:
Here, common difference (d) = -10
First term 200
1
? a
d n a a
n
) 1 (
1
? ? ?
) 10 )( 1 20 ( 200
20
? ? ? ? a
10
20
? a
4) An arithmetic sequence has a common difference equal to 10 and its 6
th
term is equal to 52. Find its
15
th
term
Solution:
We use nth term formula d n a a
n
) 1 (
1
? ? ? for the 6
th
term
) 1 6 ( 10
52
1 6
6
? ? ?
?
a a
a
50 52
1
? ?a
Therefore, 2 50 52
1
? ? ? a
Now, we know the first term and the common difference, we use the nth term formula to find the
15
th
term.
142 ) 1 15 ( 10 2
15
? ? ? ? a
Arithmetic Progressions
5) If x times the x
th
term of an AP is equal to y times its y
th
term , Prove that the (x + y)
th
term of the
AP is zero
Solution:
Let ‘a’ be the first term and‘d’ be the common difference of an AP
By the given condition,
y x
yt xt ?
? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ?
? ?
? ?
? ?
0
0 ........ 0 ) 1 (
0 ) 1 ( ) (
0 ) ( ) )( ( ) (
0 ) ( ) (
0 ) ( ) (
) ( ) (
2 2
2 2
2 2
? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
?y x
t
x d y x a
d y x a y x
d y x y x y x a y x
d y x y x a y x
d y y d x x ya xa
d y y ya d x x xa
?
6) Which term of AP 7, 14, 21,……..is 133
Solution:
Here, a = 7 , d = 14 – 7= 7
Let 133 be the
th
n term of the AP ( Where N n ? )
133 ?
n
t
133 ) 1 ( ? ? ? ? d n a
? ? N n
n
n
n
n
? ? ?
? ?
? ?
? ? ? ?
? ? ? ?
19 . .......... 19
7
133
133 7
133 7 7 7
133 7 ) 1 ( 7
Therefore, 133 is the 19
th
term of the given AP.
Arithmetic Progressions
7) Which term of AP 10, 17, 24,……..is 150
Solution:
Here, a = 10 , d =17 – 10 = 7
Let 150 be the
th
n term of the AP ( Where N n ? )
150 ?
n
t
n
t d n a ? ? ? ? ) 1 (
? ? N n
n
n
n
n
n
? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ? ? ?
21 . .......... 21
7
147
147 7
150 7 3
150 7 7 10
150 7 ) 1 ( 10
Therefore, 150 is the 21
th
term of the given AP.
8) An arithmetic sequence has its 5
th
term equal to 22 and its 15
th
term equal to 62. Find its 100
th
term
Solution:
We use the formula for 5
th
term and 15
th
term
) 1 ......( .......... 4 22
) 1 5 (
1
1 5
d a
d a a
? ?
? ? ?
) 2 ......( .......... 14 62
) 1 15 (
1
1 15
d a
d a a
? ?
? ? ?
Solving equation (1) and (2) we get value of d
Therefore d = 4
Now, plug in the value of d in equation (1) we get the value of
1
a
6
16 22
16 22
1
1
1
? ?
? ? ?
? ?
a
a
a
Now, we use nth term formula to find the 100
th
term
402
4 ) 1 100 ( 6
100
100
?
? ? ?
a
a
Arithmetic Progressions
Page 4
1) Find 6
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 6
6
6 1 ? ? ?
?
a
216 ) 1 (
5
6
? ? ? a = -216
2) Find 13
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 13
13
13 1 ? ? ?
?
a
2197 ) 1 (
12
13
? ? ? a = 2197
3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10.
Find the value of the 20
th
term.
Solution:
Here, common difference (d) = -10
First term 200
1
? a
d n a a
n
) 1 (
1
? ? ?
) 10 )( 1 20 ( 200
20
? ? ? ? a
10
20
? a
4) An arithmetic sequence has a common difference equal to 10 and its 6
th
term is equal to 52. Find its
15
th
term
Solution:
We use nth term formula d n a a
n
) 1 (
1
? ? ? for the 6
th
term
) 1 6 ( 10
52
1 6
6
? ? ?
?
a a
a
50 52
1
? ?a
Therefore, 2 50 52
1
? ? ? a
Now, we know the first term and the common difference, we use the nth term formula to find the
15
th
term.
142 ) 1 15 ( 10 2
15
? ? ? ? a
Arithmetic Progressions
5) If x times the x
th
term of an AP is equal to y times its y
th
term , Prove that the (x + y)
th
term of the
AP is zero
Solution:
Let ‘a’ be the first term and‘d’ be the common difference of an AP
By the given condition,
y x
yt xt ?
? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ?
? ?
? ?
? ?
0
0 ........ 0 ) 1 (
0 ) 1 ( ) (
0 ) ( ) )( ( ) (
0 ) ( ) (
0 ) ( ) (
) ( ) (
2 2
2 2
2 2
? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
?y x
t
x d y x a
d y x a y x
d y x y x y x a y x
d y x y x a y x
d y y d x x ya xa
d y y ya d x x xa
?
6) Which term of AP 7, 14, 21,……..is 133
Solution:
Here, a = 7 , d = 14 – 7= 7
Let 133 be the
th
n term of the AP ( Where N n ? )
133 ?
n
t
133 ) 1 ( ? ? ? ? d n a
? ? N n
n
n
n
n
? ? ?
? ?
? ?
? ? ? ?
? ? ? ?
19 . .......... 19
7
133
133 7
133 7 7 7
133 7 ) 1 ( 7
Therefore, 133 is the 19
th
term of the given AP.
Arithmetic Progressions
7) Which term of AP 10, 17, 24,……..is 150
Solution:
Here, a = 10 , d =17 – 10 = 7
Let 150 be the
th
n term of the AP ( Where N n ? )
150 ?
n
t
n
t d n a ? ? ? ? ) 1 (
? ? N n
n
n
n
n
n
? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ? ? ?
21 . .......... 21
7
147
147 7
150 7 3
150 7 7 10
150 7 ) 1 ( 10
Therefore, 150 is the 21
th
term of the given AP.
8) An arithmetic sequence has its 5
th
term equal to 22 and its 15
th
term equal to 62. Find its 100
th
term
Solution:
We use the formula for 5
th
term and 15
th
term
) 1 ......( .......... 4 22
) 1 5 (
1
1 5
d a
d a a
? ?
? ? ?
) 2 ......( .......... 14 62
) 1 15 (
1
1 15
d a
d a a
? ?
? ? ?
Solving equation (1) and (2) we get value of d
Therefore d = 4
Now, plug in the value of d in equation (1) we get the value of
1
a
6
16 22
16 22
1
1
1
? ?
? ? ?
? ?
a
a
a
Now, we use nth term formula to find the 100
th
term
402
4 ) 1 100 ( 6
100
100
?
? ? ?
a
a
Arithmetic Progressions
9) Determine p, so that
3
1
, p and p
5
2
are the three consecutive terms of an AP
Solution:
We are given
3
1
, p, p
5
2
are in AP
p p p ? ? ?
5
2
3
1
5
5 2
3
1 3 p p p ?
?
?
?
24
5
24 5
15 6 5 15
? ?
? ? ? ?
? ? ? ?
p
p
p p p
Therefore,
24
5
? p
10) Determine x, so that
7
6
, x and x
10
8
are the three consecutive terms of an AP
Solution:
We are given
7
6
, x, x
10
8
are in AP
x x x ? ? ?
10
8
7
6
10
10 8
7
6 7 x x x ?
?
?
?
7
5
84 60
70 56 60 70
? ?
? ? ? ?
? ? ? ?
x
x
x x x
Therefore,
7
5
? x
11) Which term of the progression 15 ,
5
1
14 , ,...
5
2
13 is the first negative term
Arithmetic Progressions
Page 5
1) Find 6
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 6
6
6 1 ? ? ?
?
a
216 ) 1 (
5
6
? ? ? a = -216
2) Find 13
th
term of the sequence defined by
3 1
) 1 ( n a
n
n
? ? ?
?
Solution:
Here,
3 1
) 1 ( n a
n
n
? ? ?
?
? ?
3 1 13
13
13 1 ? ? ?
?
a
2197 ) 1 (
12
13
? ? ? a = 2197
3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10.
Find the value of the 20
th
term.
Solution:
Here, common difference (d) = -10
First term 200
1
? a
d n a a
n
) 1 (
1
? ? ?
) 10 )( 1 20 ( 200
20
? ? ? ? a
10
20
? a
4) An arithmetic sequence has a common difference equal to 10 and its 6
th
term is equal to 52. Find its
15
th
term
Solution:
We use nth term formula d n a a
n
) 1 (
1
? ? ? for the 6
th
term
) 1 6 ( 10
52
1 6
6
? ? ?
?
a a
a
50 52
1
? ?a
Therefore, 2 50 52
1
? ? ? a
Now, we know the first term and the common difference, we use the nth term formula to find the
15
th
term.
142 ) 1 15 ( 10 2
15
? ? ? ? a
Arithmetic Progressions
5) If x times the x
th
term of an AP is equal to y times its y
th
term , Prove that the (x + y)
th
term of the
AP is zero
Solution:
Let ‘a’ be the first term and‘d’ be the common difference of an AP
By the given condition,
y x
yt xt ?
? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ?
? ?
? ?
? ?
0
0 ........ 0 ) 1 (
0 ) 1 ( ) (
0 ) ( ) )( ( ) (
0 ) ( ) (
0 ) ( ) (
) ( ) (
2 2
2 2
2 2
? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
?y x
t
x d y x a
d y x a y x
d y x y x y x a y x
d y x y x a y x
d y y d x x ya xa
d y y ya d x x xa
?
6) Which term of AP 7, 14, 21,……..is 133
Solution:
Here, a = 7 , d = 14 – 7= 7
Let 133 be the
th
n term of the AP ( Where N n ? )
133 ?
n
t
133 ) 1 ( ? ? ? ? d n a
? ? N n
n
n
n
n
? ? ?
? ?
? ?
? ? ? ?
? ? ? ?
19 . .......... 19
7
133
133 7
133 7 7 7
133 7 ) 1 ( 7
Therefore, 133 is the 19
th
term of the given AP.
Arithmetic Progressions
7) Which term of AP 10, 17, 24,……..is 150
Solution:
Here, a = 10 , d =17 – 10 = 7
Let 150 be the
th
n term of the AP ( Where N n ? )
150 ?
n
t
n
t d n a ? ? ? ? ) 1 (
? ? N n
n
n
n
n
n
? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ? ? ?
21 . .......... 21
7
147
147 7
150 7 3
150 7 7 10
150 7 ) 1 ( 10
Therefore, 150 is the 21
th
term of the given AP.
8) An arithmetic sequence has its 5
th
term equal to 22 and its 15
th
term equal to 62. Find its 100
th
term
Solution:
We use the formula for 5
th
term and 15
th
term
) 1 ......( .......... 4 22
) 1 5 (
1
1 5
d a
d a a
? ?
? ? ?
) 2 ......( .......... 14 62
) 1 15 (
1
1 15
d a
d a a
? ?
? ? ?
Solving equation (1) and (2) we get value of d
Therefore d = 4
Now, plug in the value of d in equation (1) we get the value of
1
a
6
16 22
16 22
1
1
1
? ?
? ? ?
? ?
a
a
a
Now, we use nth term formula to find the 100
th
term
402
4 ) 1 100 ( 6
100
100
?
? ? ?
a
a
Arithmetic Progressions
9) Determine p, so that
3
1
, p and p
5
2
are the three consecutive terms of an AP
Solution:
We are given
3
1
, p, p
5
2
are in AP
p p p ? ? ?
5
2
3
1
5
5 2
3
1 3 p p p ?
?
?
?
24
5
24 5
15 6 5 15
? ?
? ? ? ?
? ? ? ?
p
p
p p p
Therefore,
24
5
? p
10) Determine x, so that
7
6
, x and x
10
8
are the three consecutive terms of an AP
Solution:
We are given
7
6
, x, x
10
8
are in AP
x x x ? ? ?
10
8
7
6
10
10 8
7
6 7 x x x ?
?
?
?
7
5
84 60
70 56 60 70
? ?
? ? ? ?
? ? ? ?
x
x
x x x
Therefore,
7
5
? x
11) Which term of the progression 15 ,
5
1
14 , ,...
5
2
13 is the first negative term
Arithmetic Progressions
Solution:
The given expression is15,
5
1
14 , ,...
5
2
13
Here, 15
5
71
1 2
? ? ?T T
=
5
75 71 ?
=
5
4
? …………..(1)
5
71
5
67
2 3
? ? ?T T =
5
4
?
Therefore, (1) is an AP with a = 15,
5
4
? ? d
Let the nth term of the given AP be the first negative term.
Then, nth term < 0.
? 0 ?
n
T
? 0
5
4
) 1 ( 15 ?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ? n
? ? 0 4 79 ? ? ? n
79 4 ? ? n
4
3
19 ? ?n
Therefore, n = 20, i.e. 20
th
term is the first negative term in the given AP.
12) Which term of the progression 27,
5
2
26 , ,...
5
4
25 is the first negative term
Solution:
The given expression is 27 ,
5
2
26 , ,...
5
4
25
Here, 27
5
132
1 2
? ? ?T T
=
5
135 132 ?
Arithmetic Progressions
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