Chapter 9: Arithmetic Progressions Exercise 9.3

Question: 1

Find:

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

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

(iii) n^th term of the AP 13, 8, 3, -2, .......... 

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

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

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

(vii) 9^th 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.

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

10^th 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 

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

18^th term of A. P. = √2 + (18 - 1)2√2

= √2 + 17.2√2

= √2 (1+34)

= 35√2

∴ 18^th 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

n^th term of an A.P. a_n = a +(n - 1)d

= 13 + (n - 1) - 5

= 13 - 5n + 5

a_n = 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

n^th term of an A.P. a_n = a + (n - 1)d

10^th 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

n^th term = a + (n - 1)d

8^th 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

n^th term; a_n = a + (n - 1)d

11^th 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

n^th term a_n = a + (n - 1)d

9^th 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

n^th term (a_n) = a + (n - 1)d

Given n^th term a_n = 248

248 = 3+(n - 1).5

248 = -2 + 5n

5n = 250

n =250/5 = 50

50^th term is 248.

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

First term (a) = 84

Common difference (d) = a₂ - a

= 80 - 84

= – 4

n^th term (a_n) = 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

22^nd term is 0.

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

Fiat term (a) = 4

Common difference (d) = a² - a

= 9 - 4

= 5

n^th term (a_n) = a + (n - 1)d

Given n^th term is 254

4 + (n - 1)5 = 254

(n - 1)∙5 = 250

n - 1 = 250/5 = 50 

n = 51

∴  51^th term is 254. 

(iv) Given A. P

21, 42, 63, 84, ......... 

a = 21, d = a₂ - a

= 42 - 21

= 21

n^th term (a_n) = a +(n - 1)d

Given nth term = 420

21 + (n - 1)21 = 420

(n - 1)21 = 399

n - 1 = 399/21 = 19

n = 20

∴ 20^th term is 420.

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

Fiat term (a) = 121

Common difference (d) = 117 - 121

= - 4

n^th term (a) = a + (n - 1)d

Given n^th term is negative i.e., a_n < 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.

∴ 32^nd term is first negative term