RD Sharma Solutions Chapter 9 Arithmetic Progression Exercise 9.2 Class 10 Maths
Chapter Name  RD Sharma Chapter 9 Arithmetic Progression 
Book Name  RD Sharma Mathematics for Class 10 
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Related Study  NCERT Solutions for Class 10 Maths 
Exercise 9.2 Solutions
1. For the following arithmetic progressions write the first term a and the common difference d:
(i)  5, 1, 3,7, ........
(ii) 1/5, 3/5, 5/5, 7/5, .......
(iii) 0.3, 0.55, 0.80, 1.05, ......
(iv) 1.1, 3.1, 5.1, 7.1, .....
Solution
We know that if a is the first term and d is the common difference, the arithmetic progression is a, a +d, a + 2d + a + 3d, .......
(i)  5, 1, 3,7, ........
Given arithmetic series is
 5, 1, 3,7, ........
This is in the form of a, a + d, a + 2d + a + 3d, ...... by comparing these two
a = 5, a + d = 1, a + 2d = 3, a + 3d= 7, ........
First term (a) = 5
By subtracting second and first term, we get
(a + d)  (a) = d
1(5) = d
4 = d
Common difference (d) = 4
(iii) 0.3, 0.55, 0.80, 1.05, .......
Given arithmetic series,
0.3, 0.55, 0.80,1.05, .....
General arithmetic series
a, a +d, a + 2d, a + 3d,....
By comparing,
a = 0.3, a + d = 0.5, a + 2d = 0.80, a + 3d = 1.05
First term(a) = 0.3
By subtracting first term from second term. We get
d = (a + d)  (a)
d = 0.55  0.3
d = 0.25
Common difference (d) = 0.25
(iv) 1.1, 3.1, 5.1, 7.1,......
General series is
a, a + d, a + 2d, a+ 3d, .....
By comparing this two, we get
a =  1.1, a + d= 3.1, a + 2d= 5.1, a + 3d = 71
First term (a) = 1.1
Common difference (d) = (a + d)  (a)
= 3.1  (1.1)
Common difference (d) = 2
2. Write the arithmetic progressions write first term a and common difference d are as follows:
(i) a = 4, d = 3
(ii) a = 1, d = 1/2
(iii) a = 1.5, d = 0.5
Solution
We know that, if first term (a) = a and common difference = d, then the arithmetic series is, a, a+d, a+2d, a+3d, ......
(i) a = 4, d = 3
Given first term (a) = 4
Common difference(d) = 3
Then arithmetic progression is,
a, a+ d, a + 2d, a+ 3d, ......
⇒ 4, 4  3, a + 2(3), 4 + 3(3),.....
⇒ 4, 1, 2; 5, 8,.....
(ii) Given , First term(a) = 1
Common difference(d) = 1/2
Then arithmetic progression is,
⇒ a, a+d, a+ 2d, a+ 3d, .......
(iii) Given, First term(a) = 1.5
Common difference(d) = 0.5
Then arithmetic progression is
⇒ a, a + d, a +2d, a + 3d,.....
⇒ 1.5, 1.5  0.5, 1.5+2(0.5), 1.5 + 3(0.5)
⇒ 1.5, 2, 2.5, 3,.....
Then required progression is
1.5, 2, 2.5, 3,....
3. In which of the following situations, the sequence of numbers formed will form an A.P.?
(i) The cost of digging a well for the first metre is Rs 150 and rises by Rs 20 for each succeeding metre.
(ii) The amount of air present in the cylinder when a vacuum pump removes each time 1/4 of their remaining in the cylinder.
Solution
(i) Given,
Cost of digging a well for the first meter(c_{1} ) = Rs. 150.
Cost rises by Rs. 20 for each succeeding meter
Then,
Cost of digging for the second meter (c_{2} )= Rs. 150 + Rs 20 = Rs 170
Cost of digging for the third meter (c_{3} ) = Rs. 170 + Rs 20 = Rs 190
Thus, costs of digging a well for different lengths are 150, 170, 190, 210,......
Clearly, this series in in A.P.
With first term (a) = 150, common difference(d) = 20
(ii) Given,
4. Show that the sequence defined by a_{n} = 5n  7 is an A.P., find its common difference.
(ii) 0, 3, 6, 9, .......
(iii) 1, 1/4, 3/2,.......
(iv) 1, 5/6, 2/3,.......
8. Prove that no matter what the real numbers a and b are, the sequence with nth term a + nb is always an A.P. What is the common difference ?
∴ common difference (d) does not depend on nth value so, given sequence is in A.P. with (d) = b
General arithmetic progression is a, a + d, a +2d, a + 3d, .......
Common difference (d) = Second term  first term
= (d + d)  a = d (or )
= Third term  second term
= (a + 2d)  (a +d) = d
To check given sequence is in A.P. or not we use this condition.
Second term  First term = Third term  Second term
a_{1} = 3, a_{2} = 6, a_{3} = 12, a_{4} = 24
Second term  First term = 6  3= 3
Third term  Second term = 12  6 = 6
This two are not equal so given sequence is not in A.P.
In the given sequence
a_{1} = 0, a_{2} =  4, a_{3} = 8, a_{4} = 12
Second term  first term = third term  second term
a_{2}  a_{1} = a_{3}  a_{2}
4  0 = 8  (4)
4 = + 8 + 4
4 = 4
Condition is satisfied ∴ given sequence is in A.P. with common difference
In the given sequence
Condition is not satisfied
∴ given sequence not in A.P.
In the given sequence
a_{1} = 12, a_{2} = 2, a_{3} = 8, a_{4} = 18
Check the condition
a_{2}  a_{1} = a_{3}  a_{2}
2  12 = 8  2
10 = 10
∴ given sequence is in A.P. with common difference d = 10
In the given sequence
a_{1} = 3, a_{2} = 3, a_{3} = 3, a_{4} = 3
a_{2}  a_{1} = a_{3}  a_{2}
3  3 = 3  3
0 = 0
∴ given sequence is in A.P. with common difference d = 0
In the given sequence
Check the condition
a_{2}  a_{1} = a_{3}  a_{2}
P + 90  P = P + 180  P  90
90 = 180  90
90 = 90
∴ given sequence is in A.P. with common difference d = 90
In the given sequence
a_{1} = 1.0, a_{2} = 1.7, a_{3} = 2.4, a_{4} = 3.1
1.7  1.0 = 2.4  1.7
0.7 = 0.7
∴ The given sequence is in A.P with d = 0.7
In the given sequence
a_{1} = 225, a_{2} = 425, a_{3} = 625, a_{4} = 825
Check the condition
a_{2}  a_{1} = a_{3}  a_{2}
425 + 225 = 625 +425
200 = 200
∴ The given sequence is in A.P. with d = 200
In the given sequence
a_{1} = 10, a_{2} = 10 + 2^{5} , a_{3} = 10 + 2^{6} , a_{4} = 10 + 2^{7}
10 + 2^{5}  10 = 10 + 2^{6}  10 + 2^{5}
2^{5} ≠ 2^{6}  2^{5}
∴ The given sequence is not in A.P.