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Question 1:

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

$(i)2,4,8,16,........$

$(ii)2,\frac{5}{2},3,\frac{7}{2},...............$

$(iii)-1.2, -3.2,-5.2,-7.2,.............$

$(iv)-10,-6,-2,2,...............$

$(v)3,3+\sqrt{2},3+2\sqrt{2},..............$

$(vi)0.2,0.22,0.222,0.2222,............$

$(vii)0,-4,-8-12,...........$

$(viii) 1, 3, 9, 27, . . .$

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

Solution 1:

$(i)$ a₂ − a₁ = 4 − 2 = 2

a₃ − a₂= 8 − 4 = 4

$a_{2}-a_{1} eq a_{3}-a_{2}$

Therefore, the given numbers do not form an A.P.

$(ii) a_{2}-a_{1}=\frac{5}{2}-2=\frac{1}{2}$

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

$a_{2}-a_{1}= a_{3}-a_{2}$

Therefore, the given numbers form an A.P.

Common difference $d=\frac{1}{2}$

Three more terms are

$a_{4}=\frac{5}{2}+\frac{1}{2}=\frac{7}{2}$

$a_{5}=\frac{7}{2}+\frac{1}{2}=4$

$a_{6}=4+\frac{1}{2}=\frac{9}{2}$

$(iii)$ a₂ − a₁ = (−3.2) − (−1.2) = −2

a₃ − a₂= (−5.2) − (−3.2) = −2

$a_{2}-a_{1}= a_{3}-a_{2}$

Therefore, the given numbers form an A.P.

d = −2. The other terms are

$a_{4} = − 5.2 − 2=−7.2$

a₅ = − 7.2 − 2 = −9.2

a₆ = − 9.2 − 2 = −11.2

$(iv)$ a₂ − a₁ = (−6) − (−10) = 4

a₃ − a₂= (−2) − (−6) = 4

$a_{2}-a_{1}= a_{3}-a_{2}$

Therefore, the given numbers form an A.P.

d = 4 The other terms are

$a_{5}$ = 2 + 4 = 6

$a_{6}$ = 6 + 4 = 10

$a_{7}= 10+4=14$

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

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

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

$a_{2}-a_{1}= a_{3}-a_{2}$

Therefore, the given numbers form an A.P.

$d=\sqrt{2}$

Three other terms are:

$a_{5}=3+3\sqrt{2}+\sqrt{2}=3+4\sqrt{2}$

$a_{6}=3+4\sqrt{2}+\sqrt{2}=3+5\sqrt{2}$

$a_{7}=3+5\sqrt{2}+\sqrt{2}=3+6\sqrt{2}$

$(vi)$ a₂ − a₁ = 0.22 − 0.2 = 0.02

a₃ − a₂= 0.222 − 0.22 = 0.002

$a_{2}-a_{1} eq a_{3}-a_{2}$

Therefore, the given numbers do not form an A.P.

$(vii)$ a₂ − a₁ = (−4) − 0 = −4

a₃ − a₂= (−8) − (−4) = −4

$a_{2}-a_{1}= a_{3}-a_{2}$

Therefore, the given numbers form an A.P.

$(viii) a_{2}-a_{1}= \left ( -\frac{1}{2} \right )-\left ( -\frac{1}{2} \right )=0$

$a_{3}-a_{2}= \left ( -\frac{1}{2} \right )-\left ( -\frac{1}{2} \right )=0$

$a_{2}-a_{1}= a_{3}-a_{2}$

Therefore, the given numbers form an A.P.

d = 0 Three more terms are:

$a_{4}= \left ( -\frac{1}{2} \right )-o=\frac{-1}{2}$

$a_{5}= \left ( -\frac{1}{2} \right )-0=\frac{-1}{2}$

$a_{6}= \left ( -\frac{1}{2} \right )-0=\frac{-1}{2}$

$(ix)$ a₂ − a₁ = 3 − 1 = 2

a₃ − a₂= 9 − 3 = 6

$a_{2}-a_{1} eq a_{3}-a_{2}$

Therefore, the given numbers do not form an A.P.

Comments

Tasneem

Question 2:

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?

(i) The taxi fare after each km when the fare is Rs. 15 for the first km and Rs. 8 for each additional km.

(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.

(iii) The cost of digging a well after every metre of digging, when it costs Rs. 150 for the first metre and rises by ` 50 for each subsequent metre.

(iv) The amount of money in the account every year, when Rs. 10000 is deposited at compound interest at 8% per annum.

Solution 2:

(i) Given, Taxi fare for 1^st km = 15

Taxi fare for first 2 km = 15 + 8 = 23

Taxi fare for first 3 km = 23 + 8 = 31

Taxi fare for first 4 km = 31 + 8 = 39

The sums15, 23, 31, 39 forms an A.P. because every term is 8 more than the preceding term.

(ii) Let the initial volume of air in a cylinder be V litre.

In each stroke, the vacuum pump removes of air remaining in the cylinder at a time. After every stroke, only part of air will remain.

Therefore the volumes will be $V, \frac{3}{4}V,\left ( \frac{3}{4} \right )^{2}V, \left ( \frac{3}{4} \right )^{3}V....$

The adjacent terms of this series do not have the same difference between them. Therefore, this is not an A.P.

(iii) Cost of digging for first metre = 150

Cost of digging for first 2 metres = 150 + 50 = 200

Cost of digging for first 3 metres = 200 + 50 = 250

Cost of digging for first 4 metres = 250 + 50 = 300

150, 200, 250, 300 forms an A.P. because every term is 50 more than the preceding term.

(iv) We know that if Rs P is deposited at r% compound interest per annum for n years, our money will be $P\left ( 1+\frac{r}{100} \right )^{n}$

after n years.

Therefore, after every year, our money will be $10000\left ( 1+\frac{8}{100} \right ), 10000\left ( 1+\frac{8}{100} \right )^{2}, 10000\left ( 1+\frac{8}{100} \right )^{3}, 10000\left ( 1+\frac{8}{100} \right )^{4}...............$

The terms do not form an AP because consecutive sequence does not have the same difference.

Comments

Tasneem

Question 3:

Write first four terms of the AP, 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 = – 1, d = 1/2

(v) a = – 1.25, d = – 0.25

Solution 3:

i) a=10, d=10

gen.form of AP is a, a+d, a+2d,a+3d,a+4d,.....a+(n-1)d

First fourterms of AP are 10,20,30,40

ii)a=-2, d=0

First four terms of an AP are - 2,-2,-2,-2

iii)a=4, d=-3

First four terms of an AP are

=4,4-3,4+2(-3),4+3(-3)

=4,1,-2,-5 are first four terms of an AP

iv)a=-1, d=-0.25

=-1,-1-0.25,-1+2(-0.25),-1+3(-0.25)

=-1,-1.25,-0.50,-0.25 are first four terms of an AP

v) a=-1.25, d=-0.25

=-1.25,-1.25-0.25,-1.25+2(-0.25),-1.25+3(-0.25)

=-1.25,-1.50,-1.75,-2.0 are first four terms of an AP

Comments

Sana

Question 4:

For the following APs, write the first term and the common difference:

$(i)3,1,-1,-3,.........$

$(ii)-5,-1,3,7,..................$

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

$(iv)0.6,1.7,2.8,3.9.,..............$

Solution 4:

Formula to find the common difference =second term-first term

(i) 3, 1, −1, −3 …

Here, first term, a = 3

Common difference, d = 1 − 3 = −2

(ii) −5, −1, 3, 7 …

a = −5

d = (−1) − (−5) = − 1 + 5 = 4

(iii)

(iv) 0.6, 1.7, 2.8, 3.9 …

a = 0.6

d = 1.7 − 0.6

= 1.1

Comments

Tasneem

All Exercises - Chapter 5 - Arithmetic Progression

Exercise 5.1

Exercise 5.2

Exercise 5.3

Exercise 5.4

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Learn Exercise 5.1 with Free Lessons & Tips

All Exercises - Chapter 5 - Arithmetic Progression

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