ML Aggarwal Solution Class 10 Chapter 9 Arithmetic and Geometric Progressions Exercise 9.1

 Exercise 9.1

Question 1

For the following A.P.s, write the first term a and the common difference d:

(i) 3, 1, – 1, – 3, …

(ii) $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots .$

(iii) – 3.2, – 3, – 2.8, – 2.6, …

Sol :

(i) 3, 1, -1, -3, …

Here first term (a) = 3

and the common difference (d)

= 1 – 3 = -2,

– 1 – 1 = -2,…

= -2

(ii) $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots$

Here first common term $(a)=\frac{1}{3}$

and common difference $(d)=$

$\frac{5}{3}-\frac{1}{3}=\frac{4}{3}, \frac{9}{3}-\frac{5}{3}=\frac{4}{3}, \ldots$

$=\frac{4}{3}$

(iii) -3.2,-3,-2.8,-2.6, ....

Here first term (a)=-3.2

and common difference (d)

=-3-(-3.2)=-3+3.2=0.2

=(d)=0.2

Question 2

Write first four terms of the A.P., when the first term a and the common difference d are given as follows :

(i) a = 10, d = 10

(ii) a = – 2, d = 0

(iii) a = 4, d = – 3

(iv) $a=\frac{1}{2}, d=-\frac{1}{2}$

Sol :

(i) a = 10, d = 10

∴ A.P. = 10, 20, 30, 40, …

(ii) a = -2, d = 0

∴ A.P. = -2, -2, -2, -2, ….

(iii) $a=4, d=-3$

$\therefore$ A.P. $=4,1,-2,-5, \ldots$

(iv) $a=\frac{1}{2}, d=-\frac{1}{6}$

A.P. is $\frac{1}{2},\left(\frac{1}{2}-\frac{1}{6}\right)=\frac{2}{6}$

$=\frac{2}{6}-\frac{1}{6}=\frac{1}{6}, \ldots$

A.P. $=\frac{1}{2}, \frac{2}{6}, \frac{1}{6}, 0, \ldots$

$=\frac{1}{2}, \frac{1}{3}, \frac{1}{6}, 0, \ldots$

Question 3

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms :

(i) 4, 10, 16, 22,…

(ii) – 2, 2, – 2, 2,…..

(iii) 2, 4, 8, 16,….

(iv) $2, \frac{5}{2}, 3, \frac{7}{2}, \ldots \ldots$

(v) – 10, – 6, – 2, 2,….

(vi) 1², 3², 5², 7²,….

(vii) 1, 3, 9, 27,….

(viii) √2, √8, √18, √32,….

(ix) 3, 3 + √2, 3 + √2, 3 + 3√2,…..

(x) √3, √6, √9, √12,……

(xi) a, 2a, 3a, 4a,…….

(xii) a, 2a + 1, 3a + 2, 4a + 3,….

Sol :

(i) 4, 10, 16, 22,…

Here a = 4, d = 10 – 4 = 6, 16 – 10 = 6, 22 – 16 = 6

∵ common difference is same

∵ It is in A.P

and next three terms are 28, 34, 40

(ii) $-2,2,-2,2, \ldots$

Here, $a=-2$

$d=2-(-2)=2+2=4$

$-2-2=-4$

$2-(-2)=4$

$\because$ Common difference is not same.

$\therefore$ It is not an A.P.

(iii) 2,4,8,16,...

Here, $a=2$

d=4-2=2,8-4=4,16-8=8

$\therefore$ Common difference is not same.

$\therefore$ It is not an A.P.

(iv) $2, \frac{5}{2}, 3, \frac{7}{2}, \ldots$

Here a=2

$d=\frac{5}{2}-2=\frac{1}{2}$

$3-\frac{5}{2}=\frac{1}{2}$

$\frac{7}{2}-3=\frac{1}{2}$

$\because$ Common difference is same.

$\therefore$ It is an A.P.

and next three terms are $4, \frac{9}{2}, 5$

(v) $-10,-6,-2,2, \ldots$

Here, first term $(a)=-10$

$d=-6-(-10)=-6+10=4$

$-2-(-6)=-2+6=4$

$2-(-2)=2+2=4$

∴Common difference is same.

∴It is an A.P.

and next three terms are 6,10,14,20

(vi)

$1^{2}, 3^{2}, 5^{2}, 7^{2}, \ldots$

=1,9,25,49, ...

Here, first term $(a)=1^{2}=1$

d=9-1=8

25-9=16

49-25=24

∵Common difference is not same.

∴It is not an A.P.

(vii) 1,3,9,27,...

Here , first term (a)=1

d=3-1=2

9-3=6

27-9=18

∵Common difference is not same.

∴It is not an A.P.

(viii) $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots$

⇒$\sqrt{2}, 2 \sqrt{2}, 3 \sqrt{2}, 4 \sqrt{2}, \ldots$

Here, first term $(a)=\sqrt{2}$

and common difference $(d)$

$=2 \sqrt{2}-\sqrt{2}=\sqrt{2}$

$=3 \sqrt{2}-2 \sqrt{2}=\sqrt{2}$

$=4 \sqrt{2}-3 \sqrt{2}=\sqrt{2}$

$\because$ The common difference is same

$\therefore$ It is an A.P.

$\sqrt{50}, \sqrt{72}, \sqrt{98}, \ldots$

(ix) $3,3+\sqrt{2}, 3+2 \sqrt{2}, 3+3 \sqrt{2}, \ldots$

Here, first term $(a)=3$

and $d=3+\sqrt{2}-2=\sqrt{2}$

$3+2 \sqrt{2}-3-\sqrt{2}=\sqrt{2}$

$3+3 \sqrt{2}-3+2 \sqrt{2}=\sqrt{2}$

∵Common difference is same.

∴It is an A.P.

and next three terms are

$3+4 \sqrt{2}, 3+5 \sqrt{2}, 3+6 \sqrt{2}, \ldots$

(x) $\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots$

Here, $a=\sqrt{3}$

$d=\sqrt{6}-\sqrt{3}=\sqrt{3} \times \sqrt{2}-\sqrt{3}$

$=\sqrt{3}(\sqrt{2}-1)$

$=\sqrt{9}-\sqrt{6}=3-\sqrt{2} \sqrt{3}=\sqrt{3}(\sqrt{3}-\sqrt{2})$

∴Common difference is not same.

∴It is not an A.P.

(xi) a, 2 a, 3 a, 4 a, ...

Here first term (a)=a

Common difference (d)=2 a-a=a 

3a-2a=a

4a-3 a=a

∵The common difference is same.

∴It is an A.P.

and next three terms are

5a, 6a, 7a

(xii) a,2a+1, 3a+2, 4a+3... 

Here, first term (a)=a

and common difference (d)

=2a+1-a=a+1

3a+2-2a-1=a+1

4a+3-3a-2=a+1

∵The common difference is same.

∴It is an A.P.

and next three terms are

5a+4, 6a+5, 7a+6...