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Chapter 9: Arithmetic Progressions Exercise 9.3

Question: 1

Find:

(i) 10th tent of the AP 1, 4, 7, 10....

(ii) 18th term of the AP √2, 3√2, 5√2, ……. 

(iii) nth term of the AP 13, 8, 3, -2, .......... 

(iv) 10th term of the AP -40, -15, 10, 35, ............. 

(v) 8th term of the AP 11, 104, 91, 78, ............... 

(vi) 11th tenor of the AP 10.0, 10.5, 11.0, 11.2, .............. 

(vii) 9th term of the AP 3/4, 5/4, 7/4 + 9/4, ...........

Solution:

Given A.P. is 1, 4, 7, 10, .......... 

First term (a) = 1

Common difference (d) = second than first term

= 4 - 1 = 3.

nth term in an A.p = a + (n - 1)d

10th term in an 1 + (10 - 1)3

= 1 + 9.3

= 1 + 27

= 28

(ii) Given A.P. is

√2, 3√2, 5√2, …….

Fiat term (a) = √2 

Common difference = Second term – First term

= 3√2 - √2

d = 2√2 

nth term in an A. P. = a + (n - 1)d

18th term of A. P. = √2 + (18 - 1)2√2

= √2 + 17.2√2

= √2 (1+34)

= 35√2

∴ 18th term of A. P. is 35√2

(iii) Given A. P. is

13, 8, 3, - 2,  ............ 

First term (a) = 13

Common difference (d) = Second term first term

= 8 - 13 = – 5

nth term of an A.P. an = a +(n - 1)d

= 13 + (n - 1) - 5

= 13 - 5n + 5

an = 18 - 5n

(iv) Given A. P. is

- 40, -15, 10, 35, .......... 

First term (a) = -40

Common difference (d) = Second term - fast term

= -15 - (- 40)

= 40 - 15

= 25

nth term of an A.P. an = a + (n - 1)d

10th term of A. P. a10 = -40 + (10 - 1)25

= – 40 + 9.25

= – 40 + 225

= 185

(v) Given sequence is 117, 104, 91, 78, ............. 

First learn can = 117

Common difference (d) = Second term - first term

= 104 - 117

= – 13

nth term = a + (n - 1)d

8th term = a + (8 - 1)d

= 117 + 7(-13)

= 117 - 91

= 26

(vi) Given A. P is

10.0, 10.5, 11.0, 11.5, 

First term (a) = 10.0

Common difference (d) = Second term - first term

= 10.5 - 10.0 = 0.5

nth term; an = a + (n - 1)d

11th term a11 = 10.0 + (11 - 1)0.5

= 10.0 + 10 x 0.5

= 10.0 + 5

=15.0

(vii) Given A. P is

3/4,  5/4, 7/4, 9/4, ............

First term (a) = 3/4

Common difference (d) = Second term - first term

= 5/4 - 3/4

= 2/4

nth term an = a + (n - 1)d

9th term a9 = a + (9 - 1)d

 

Question: 2

(i) Which term of the AP 3, 8, 13, .... is 248?

(ii) Which term of the AP 84, 80, 76, ... is 0?

(iii) Which term of the AP 4. 9, 14, .... is 254?

(iv) Which term of the AP 21. 42, 63, 84, ... is 420?

(v) Which term of the AP 121, 117. 113, ... is its first negative term?

Solution:

(i) Given A.P. is 3, 8, 13, ...........

First term (a) = 3

Common difference (d) = Second term - first term

= 8 - 3

= 5

nth term (an) = a + (n - 1)d

Given nth term an = 248

248 = 3+(n - 1).5

248 = -2 + 5n

5n = 250

n =250/5 = 50

50th term is 248.

(ii) Given A. P is 84, 80, 76, ............ 

First term (a) = 84

Common difference (d) = a2 - a

= 80 - 84

= – 4

nth term (an) = a +(n - 1)d

Given nth term is 0

0 = 84 + (n - 1) - 4

84 = +4(n -  1)

n - 1 = 84/4 = 21

n = 21 + 1 = 22

22nd term is 0.

(iii) Given A. P 4, 9, 14, ............ 

Fiat term (a) = 4

Common difference (d) = a2 - a

= 9 - 4

= 5

nth term (an) = a + (n - 1)d

Given nth term is 254

4 + (n - 1)5 = 254

(n - 1)∙5 = 250

n - 1 = 250/5 = 50 

n = 51

∴  51th term is 254. 

(iv) Given A. P

21, 42, 63, 84, ......... 

a = 21, d = a2 - a

= 42 - 21

= 21

nth term (an) = a +(n - 1)d

Given nth term = 420

21 + (n - 1)21 = 420

(n - 1)21 = 399

n - 1 = 399/21 = 19

n = 20

∴ 20th term is 420.

(v) Given A.P is 121, 117, 113, ........... 

Fiat term (a) = 121

Common difference (d) = 117 - 121

= - 4

nth term (a) = a + (n - 1)d

Given nth term is negative i.e., an < 0

121 + (n - 1) - 4 < 0

121 + 4 - 4n < 0

125 - 4n < 0

4n > 125

n > 125/4

n > 31.25

The integer which comes after 31.25 is 32.

∴ 32nd term is first negative term


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