Page 1
1. Show that each of the progressions given below is an AP. Find the first term, common
difference and next term of each.
(i)
(ii) 11, 6, 1,
(iii)
5 2 1
1, , , ,..........
6 3 2
(iv) 2, 8, 18, 32,..........
(v) 20, 45, 80, 125,.........
Sol:
(i) The given progression 9
Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 9
Common difference =6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1,
Clearly,6 11 = 1 6 = 4 1 = 5 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 11
Common difference = 5
Next term of the 4 5 9 A P
(iii) The given progression
5 2 1
1, , , ,..........
6 3 2
Clearly,
5 2 5 1 2 1
1
6 3 6 2 3 6
(Constant)
Thus, each term differs from its preceding term by
1
.
6
So, the given progression is an
AP.
First term = 1
Common difference =
1
6
Next tern of the
1 1 2 1
2 6 6 3
A P
Page 2
1. Show that each of the progressions given below is an AP. Find the first term, common
difference and next term of each.
(i)
(ii) 11, 6, 1,
(iii)
5 2 1
1, , , ,..........
6 3 2
(iv) 2, 8, 18, 32,..........
(v) 20, 45, 80, 125,.........
Sol:
(i) The given progression 9
Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 9
Common difference =6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1,
Clearly,6 11 = 1 6 = 4 1 = 5 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 11
Common difference = 5
Next term of the 4 5 9 A P
(iii) The given progression
5 2 1
1, , , ,..........
6 3 2
Clearly,
5 2 5 1 2 1
1
6 3 6 2 3 6
(Constant)
Thus, each term differs from its preceding term by
1
.
6
So, the given progression is an
AP.
First term = 1
Common difference =
1
6
Next tern of the
1 1 2 1
2 6 6 3
A P
(iv) The given progression 2, 8, 18, 32,..........
This sequence can be written as 2, 2 2,3 2,4 2,........
Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant)
Thus, each term differs from its preceding term by 2, So, the given progression is
an AP.
First term 2
Common difference 2
Next tern of the 4 2 2 5 2 50 A P
(v) This given progression 20, 45, 80, 125,.........
This sequence can be re-written as 2 5,3 5, 4 5,5 5,...........
Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant)
Thus, each term differs from its preceding term by 5.So, the given progression is
an AP.
First term 2 5 20
Common difference 5
Next term of the 5 5 5 6 5 180 A P
2. Find:
(i) the 20
th
term of the AP 9,13,17,21,..........
(ii) the 35
th
term of AP 20,17,14,11,..........
(iii) the 18
th
term of the AP 2, 18, 50, 98,...........
(iv) the 9
th
term of the AP
3 5 7 9
, , , ,.........
4 4 4 4
(v) the 15the term of the AP 40, 15,10,35,.........
Sol:
(i) The given A P is 9,13,17,21,..........
First term, 9 a
Common difference, 13 9 4 d
t h
n term of the AP, 1 9 1 4
n
a a n d n
20 t h term of the AP,
20
9 20 1 4 9 76 85 a
(ii) The given AP is 20,17,14,11,..........
First term, 20 a
Common difference, 17 20 3 d
t h
n term of the AP, 1 20 1 3
n
a a n d n
Page 3
1. Show that each of the progressions given below is an AP. Find the first term, common
difference and next term of each.
(i)
(ii) 11, 6, 1,
(iii)
5 2 1
1, , , ,..........
6 3 2
(iv) 2, 8, 18, 32,..........
(v) 20, 45, 80, 125,.........
Sol:
(i) The given progression 9
Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 9
Common difference =6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1,
Clearly,6 11 = 1 6 = 4 1 = 5 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 11
Common difference = 5
Next term of the 4 5 9 A P
(iii) The given progression
5 2 1
1, , , ,..........
6 3 2
Clearly,
5 2 5 1 2 1
1
6 3 6 2 3 6
(Constant)
Thus, each term differs from its preceding term by
1
.
6
So, the given progression is an
AP.
First term = 1
Common difference =
1
6
Next tern of the
1 1 2 1
2 6 6 3
A P
(iv) The given progression 2, 8, 18, 32,..........
This sequence can be written as 2, 2 2,3 2,4 2,........
Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant)
Thus, each term differs from its preceding term by 2, So, the given progression is
an AP.
First term 2
Common difference 2
Next tern of the 4 2 2 5 2 50 A P
(v) This given progression 20, 45, 80, 125,.........
This sequence can be re-written as 2 5,3 5, 4 5,5 5,...........
Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant)
Thus, each term differs from its preceding term by 5.So, the given progression is
an AP.
First term 2 5 20
Common difference 5
Next term of the 5 5 5 6 5 180 A P
2. Find:
(i) the 20
th
term of the AP 9,13,17,21,..........
(ii) the 35
th
term of AP 20,17,14,11,..........
(iii) the 18
th
term of the AP 2, 18, 50, 98,...........
(iv) the 9
th
term of the AP
3 5 7 9
, , , ,.........
4 4 4 4
(v) the 15the term of the AP 40, 15,10,35,.........
Sol:
(i) The given A P is 9,13,17,21,..........
First term, 9 a
Common difference, 13 9 4 d
t h
n term of the AP, 1 9 1 4
n
a a n d n
20 t h term of the AP,
20
9 20 1 4 9 76 85 a
(ii) The given AP is 20,17,14,11,..........
First term, 20 a
Common difference, 17 20 3 d
t h
n term of the AP, 1 20 1 3
n
a a n d n
35 t h term of the AP,
35
20 35 1 3 20 102 82 a
(iii) The given AP is 2, 18, 50, 98,...........
This can be re-written as 2,3 2,5 2,7 2,.........
First term, 2 a
Common difference, 3 2 2 2 2 d
t h
n term of the AP, 1 2 1 2 2
n
a a n d n
18 t h term of the AP,
18
2 18 1 2 2 2 34 2 35 2 2450 a
(iv) The given AP is
3 5 7 9
, , , ,.........
4 4 4 4
First term,
3
4
a
Common difference,
5 3 2 1
4 4 4 2
d
t h
n term of the AP,
3 1
1 1
4 2
n
a a n d n
9
t h
t e r m of the AP,
9
3 1 3 19
9 1 4
4 2 4 4
a
(v) The given AP is 40, 15,10,35,.........
First term, 40 a
Common difference, 15 40 25 d
t h
n term of the AP, 1 40 1 25
n
a a n d n
15 t h term of the AP,
15
40 15 1 25 40 350 310 a
3. Find the 37
th
term of the AP
3 1 1
6,7 ,9 ,11 ,.........
4 2 4
Sol:
The given AP is
3 1 1
6,7 ,9 ,11 ,.........
4 2 4
First term, 6 a and common difference,
3 31 31 24 7
7 6 6
4 4 4 4
d
Now,
37
37 1 36 T a d a d
7
6 36 6 63 69
4
37
t h
term = 69
Page 4
1. Show that each of the progressions given below is an AP. Find the first term, common
difference and next term of each.
(i)
(ii) 11, 6, 1,
(iii)
5 2 1
1, , , ,..........
6 3 2
(iv) 2, 8, 18, 32,..........
(v) 20, 45, 80, 125,.........
Sol:
(i) The given progression 9
Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 9
Common difference =6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1,
Clearly,6 11 = 1 6 = 4 1 = 5 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 11
Common difference = 5
Next term of the 4 5 9 A P
(iii) The given progression
5 2 1
1, , , ,..........
6 3 2
Clearly,
5 2 5 1 2 1
1
6 3 6 2 3 6
(Constant)
Thus, each term differs from its preceding term by
1
.
6
So, the given progression is an
AP.
First term = 1
Common difference =
1
6
Next tern of the
1 1 2 1
2 6 6 3
A P
(iv) The given progression 2, 8, 18, 32,..........
This sequence can be written as 2, 2 2,3 2,4 2,........
Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant)
Thus, each term differs from its preceding term by 2, So, the given progression is
an AP.
First term 2
Common difference 2
Next tern of the 4 2 2 5 2 50 A P
(v) This given progression 20, 45, 80, 125,.........
This sequence can be re-written as 2 5,3 5, 4 5,5 5,...........
Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant)
Thus, each term differs from its preceding term by 5.So, the given progression is
an AP.
First term 2 5 20
Common difference 5
Next term of the 5 5 5 6 5 180 A P
2. Find:
(i) the 20
th
term of the AP 9,13,17,21,..........
(ii) the 35
th
term of AP 20,17,14,11,..........
(iii) the 18
th
term of the AP 2, 18, 50, 98,...........
(iv) the 9
th
term of the AP
3 5 7 9
, , , ,.........
4 4 4 4
(v) the 15the term of the AP 40, 15,10,35,.........
Sol:
(i) The given A P is 9,13,17,21,..........
First term, 9 a
Common difference, 13 9 4 d
t h
n term of the AP, 1 9 1 4
n
a a n d n
20 t h term of the AP,
20
9 20 1 4 9 76 85 a
(ii) The given AP is 20,17,14,11,..........
First term, 20 a
Common difference, 17 20 3 d
t h
n term of the AP, 1 20 1 3
n
a a n d n
35 t h term of the AP,
35
20 35 1 3 20 102 82 a
(iii) The given AP is 2, 18, 50, 98,...........
This can be re-written as 2,3 2,5 2,7 2,.........
First term, 2 a
Common difference, 3 2 2 2 2 d
t h
n term of the AP, 1 2 1 2 2
n
a a n d n
18 t h term of the AP,
18
2 18 1 2 2 2 34 2 35 2 2450 a
(iv) The given AP is
3 5 7 9
, , , ,.........
4 4 4 4
First term,
3
4
a
Common difference,
5 3 2 1
4 4 4 2
d
t h
n term of the AP,
3 1
1 1
4 2
n
a a n d n
9
t h
t e r m of the AP,
9
3 1 3 19
9 1 4
4 2 4 4
a
(v) The given AP is 40, 15,10,35,.........
First term, 40 a
Common difference, 15 40 25 d
t h
n term of the AP, 1 40 1 25
n
a a n d n
15 t h term of the AP,
15
40 15 1 25 40 350 310 a
3. Find the 37
th
term of the AP
3 1 1
6,7 ,9 ,11 ,.........
4 2 4
Sol:
The given AP is
3 1 1
6,7 ,9 ,11 ,.........
4 2 4
First term, 6 a and common difference,
3 31 31 24 7
7 6 6
4 4 4 4
d
Now,
37
37 1 36 T a d a d
7
6 36 6 63 69
4
37
t h
term = 69
4. Find the 25
th
term of the AP
1 1
5,4 , 4,3 ,3,........
2 2
Sol:
The given AP is
1 1
5,4 , 4,3 ,3,........
2 2
First term = 5
Common difference
1 9 9 10 1
4 5 5
2 2 2 2
1
5
2
a a n d d
Now,
25
25 1 24 T a d a d
1
5 24 5 12 7
2
25
t h
term 7
5. Find the nth term of each of the following Aps:
(ii) 16, 9, 2, -
Sol:
(i) (6n 1)
(ii) (23 7n)
6. If the nth term of a progression is (4n 10) show that it is an AP. Find its
(i) first term, (ii) common difference (iii) 16 the term.
Sol:
4 10
n n
T [Given]
1
4 1 10 6 T
2
3
4
4 2 10 2
4 3 10 2
4 4 10 6
T
T
T
Clearly, 2 6 2 2 6 2 4 (Constant)
So, the terms 6, 2,2,6,...... forms an AP.
Thus we have
(i) First term 6
(ii) Common difference = 4
(iii)
16
1 15 6 15 4 54 T a n d a d
Page 5
1. Show that each of the progressions given below is an AP. Find the first term, common
difference and next term of each.
(i)
(ii) 11, 6, 1,
(iii)
5 2 1
1, , , ,..........
6 3 2
(iv) 2, 8, 18, 32,..........
(v) 20, 45, 80, 125,.........
Sol:
(i) The given progression 9
Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 9
Common difference =6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1,
Clearly,6 11 = 1 6 = 4 1 = 5 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an
AP.
First term = 11
Common difference = 5
Next term of the 4 5 9 A P
(iii) The given progression
5 2 1
1, , , ,..........
6 3 2
Clearly,
5 2 5 1 2 1
1
6 3 6 2 3 6
(Constant)
Thus, each term differs from its preceding term by
1
.
6
So, the given progression is an
AP.
First term = 1
Common difference =
1
6
Next tern of the
1 1 2 1
2 6 6 3
A P
(iv) The given progression 2, 8, 18, 32,..........
This sequence can be written as 2, 2 2,3 2,4 2,........
Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant)
Thus, each term differs from its preceding term by 2, So, the given progression is
an AP.
First term 2
Common difference 2
Next tern of the 4 2 2 5 2 50 A P
(v) This given progression 20, 45, 80, 125,.........
This sequence can be re-written as 2 5,3 5, 4 5,5 5,...........
Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant)
Thus, each term differs from its preceding term by 5.So, the given progression is
an AP.
First term 2 5 20
Common difference 5
Next term of the 5 5 5 6 5 180 A P
2. Find:
(i) the 20
th
term of the AP 9,13,17,21,..........
(ii) the 35
th
term of AP 20,17,14,11,..........
(iii) the 18
th
term of the AP 2, 18, 50, 98,...........
(iv) the 9
th
term of the AP
3 5 7 9
, , , ,.........
4 4 4 4
(v) the 15the term of the AP 40, 15,10,35,.........
Sol:
(i) The given A P is 9,13,17,21,..........
First term, 9 a
Common difference, 13 9 4 d
t h
n term of the AP, 1 9 1 4
n
a a n d n
20 t h term of the AP,
20
9 20 1 4 9 76 85 a
(ii) The given AP is 20,17,14,11,..........
First term, 20 a
Common difference, 17 20 3 d
t h
n term of the AP, 1 20 1 3
n
a a n d n
35 t h term of the AP,
35
20 35 1 3 20 102 82 a
(iii) The given AP is 2, 18, 50, 98,...........
This can be re-written as 2,3 2,5 2,7 2,.........
First term, 2 a
Common difference, 3 2 2 2 2 d
t h
n term of the AP, 1 2 1 2 2
n
a a n d n
18 t h term of the AP,
18
2 18 1 2 2 2 34 2 35 2 2450 a
(iv) The given AP is
3 5 7 9
, , , ,.........
4 4 4 4
First term,
3
4
a
Common difference,
5 3 2 1
4 4 4 2
d
t h
n term of the AP,
3 1
1 1
4 2
n
a a n d n
9
t h
t e r m of the AP,
9
3 1 3 19
9 1 4
4 2 4 4
a
(v) The given AP is 40, 15,10,35,.........
First term, 40 a
Common difference, 15 40 25 d
t h
n term of the AP, 1 40 1 25
n
a a n d n
15 t h term of the AP,
15
40 15 1 25 40 350 310 a
3. Find the 37
th
term of the AP
3 1 1
6,7 ,9 ,11 ,.........
4 2 4
Sol:
The given AP is
3 1 1
6,7 ,9 ,11 ,.........
4 2 4
First term, 6 a and common difference,
3 31 31 24 7
7 6 6
4 4 4 4
d
Now,
37
37 1 36 T a d a d
7
6 36 6 63 69
4
37
t h
term = 69
4. Find the 25
th
term of the AP
1 1
5,4 , 4,3 ,3,........
2 2
Sol:
The given AP is
1 1
5,4 , 4,3 ,3,........
2 2
First term = 5
Common difference
1 9 9 10 1
4 5 5
2 2 2 2
1
5
2
a a n d d
Now,
25
25 1 24 T a d a d
1
5 24 5 12 7
2
25
t h
term 7
5. Find the nth term of each of the following Aps:
(ii) 16, 9, 2, -
Sol:
(i) (6n 1)
(ii) (23 7n)
6. If the nth term of a progression is (4n 10) show that it is an AP. Find its
(i) first term, (ii) common difference (iii) 16 the term.
Sol:
4 10
n n
T [Given]
1
4 1 10 6 T
2
3
4
4 2 10 2
4 3 10 2
4 4 10 6
T
T
T
Clearly, 2 6 2 2 6 2 4 (Constant)
So, the terms 6, 2,2,6,...... forms an AP.
Thus we have
(i) First term 6
(ii) Common difference = 4
(iii)
16
1 15 6 15 4 54 T a n d a d
7.
Sol:
In the given AP, 6 a and 10 6 4 d
Suppose that there are n terms in the given AP.
Then, 174
n
T
1 174 a n d
6 1 4 174
2 4 174
4 172
43
n
n
n
n
Hence, there are 43 terms in the given AP.
8. How many terms are there in the AP 41, 3
Sol:
In the given AP, 41 38 41 3 a a n d d
Suppose that there are n terms in the given AP.
Then 8
n
T
1 8
41 1 3 8
44 3 8
3 36
12
a n d
n
n
n
n
Hence, there are 12 terms in the given AP.
9. How many terms are there in the AP
1
18,15 ,13,....., 47.
2
?
Sol:
The given AP is
1
18,15 ,13,....., 47.
2
First term, 18 a
Common difference,
1 31 31 36 5
15 18 18
2 2 2 2
d
Suppose there re n terms in the given AP. Then,
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