Exercise - 1
Q.1 Choose the correct option and give reasons:
(i) First term and common difference of given arithmetic progression are:-
3/2, 1/2, -1/2, -3/2 .......
(a) 1/2, -1/2
(b) 3/2, -1
(c) 3/2, -1 /2
(d) -3/2, -1
Ans. (d) -3/2, -1
First term = 3/2
Common Difference = second term - first term
= (1/2) - ( 3/2)
= -1
(ii) If first term is -2 and common difference is -2 for an arithmetic progression, then fourth term is _________.
(a) 0 (b) -2 (c) -4 (d) -8
Ans. (d) -8
first term is -2
common difference is -2
Fourth term = a + (4 -1)d
= (-2) + 3 x (-2)
= -2 - 6
= - 8
(iii) 15th term in the arithmetic progression 7, 13, 19, ..... is
(a) 91 (b) 97 (c) 112 (d) 90
Ans. (a) 91
First term (a) = 7
Common difference (d) = 13 - 7 = 6
15th term = a + (15 - 1)d
= 7 + 14 x 6
= 7 + 84
= 91
(iv) First term is 4 and common difference is -4 of an arithmetic progression, then nth term is:-
(a) 8 - 2n (b) 4 - 2n (c) 8 - 4n (d) 8 - 8n
Ans. (c) 8 - 4n
nth term = a + (n -1)d
= 4 + (n - 1)(-4)
= 4 - 4n + 4
= 8 - 4n
(v) 78 is which term of arithmetic progression 3, 8, 13, 18, .....
(a) 15th (b) 16th (c) 17th (d) 18th
Ans. (b) 16th
Last term = a + (n - 1)d
78 = 3 + (n - 1)5
75 = (n -1)5
15 = n - 1
n = 16
Q.2 Which is an arithmetic progression from the following progressions, give reasons:
(a) a, a2, a3,a4....
Ans. a1 = a, a2 = a2, a3
= a3, a4 = a4
a2 – a1 = a2 – a = a(a-1)
a3 – a2 = a3 – a2 = a2(a
– 1)
a2 – a1 ≠ a3 – a2
There is no common difference.
Hence given sequence is not an arithmetic progression.
(b) ........................
Ans. a1 = , a2 =
, a3 =
, a4 =
a2 – a1 = –
=2
-
=
(2-1) =
a3 – a2 = –
= 3
- 2
=
a2 – a1 = a3 – a2
Here, we get the common difference =
Hence given sequence is an arithmetic progression.
(c)
Ans. a1 = , a2 =
, a3 =
, a4 =
a2 – a1 = –
= 9 - 1 = 8
a3 – a2 = -
= 25 - 9 = 16
a2 – a1 ≠ a3 – a2
There is no common difference.
Hence given sequence is not an arithmetic progression
(d) 0,-4, -8, -12, ........
Ans. a1 = 0, a2 = -4, a3 = -8, a4 = -12
a2 – a1 = -4 - 0 = -4
a3 – a2 = -8 - (-4) = -8 + 4 = -4
a2 – a1 = a3 – a2
Here, we get the common difference = -4
Hence given sequence is an arithmetic progression
(e) 16, ,
,23,......................
Ans. a1 = 16, a2 =, a3 =
, a4 = 23
a2 – a1 = - 16 = 5/2
a3 – a2 = -
= 5/2
a2 – a1 = a3 – a2
Here, we get the common difference =5/2
Hence given sequence is an arithmetic progression
Q.3 Find the 10th term of the arithmetic progression 9, 5, 1, -3, .....
Ans.
First term(a) = 9
Common difference(d) = 5 - 9 = -4
10th term a10 = a + 9d
= 9 + 9(-4)
= 9 -36
= -27
Q.4 Find the 40th term of arithmetic progression 100, 70, 40……..
Ans.
First term (a) = 100
Common difference (d) = 70 - 100 = -30
40th term a40 = a + (40 - 1) d
= a + 39d
= 100 + 39 (-30)
= 100 - (1170)
= 1070
Q.5 Find the nth term of the arithmetic progression 1/9, 4/9, 7/9 ......
Ans.
First term (a) = 1/9
Common difference (d) = (4/9) - (1/9) = 3/9 = 1/3
nth term an = a + (n - 1) d
= (1/9) + (n - 1)(1/3)
= (1/9) + (n/3) - (1/3)
= (-2/9) + (n/3)
= (3n - 2)/9
Q.6 Find the mth term of arithmetic progression 950, 900, 850, .....
Ans.
First term (a) = 950
Common difference (d) = 900 - 950 = -50
nth term am = a + (m - 1) d
= 950 + (m - 1)(-50)
= 1000 - 50m
Q.7 Last term of the arithmetic progression 8, 15, 22, ..... is 218. Find the number of terms.
Ans.
Given, First term(a) = 8
Common difference(d) = 15 - 8 = 7
Last term = a + (n - 1)d
218 = 8 + (n - 1) 7
210 = (n -1)7
30 = n - 1
n = 31
Q.8 Which term is 0 of the AP 27, 24, 21, .....?
Ans.
Given
First term = 27
Common difference = 24 - 27 = -3
0 = 27 + (n - 1)(-3)
-27 = (n -1) (-3)
(-27)/(-3) = n - 1
9 = n - 1
n = 10
so, 10th term of the AP is 0.