Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 5 Arithmetic Progressions Ex 5.3 Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Ex 5.3

Question 1.

Find the sum of the following APs:

(i) 2, 7, 12, … to 10 terms.

(ii) – 37, – 33, – 29, ………….. to 12 terms.

(iii) 0.6, 1.7, 2.8, … to 100 terms.

(iv) \(\frac{1}{5}\), \(\frac{1}{12}\), \(\frac{1}{10}\), ………… to 11 terms.

Solution:

(i) Given AP. is 2, 7, 12, …

Here a = 2, d = 7 – 2 = 5 and n = 10

S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

∴ S_{10} = \(\frac{10}{2}\) [2 × 2 + (10 – 1)5]

= 5 [4 + 45] = 245.

(ii) Given A.P. is – 37, – 33, – 29…

Here a = -37, d = – 33 + 37 = 4 and n = 12

S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

∴ S_{12} = [2(-37) + (12 – 1)4]

= 6 [- 74 + 44] = – 180.

(iii) Given A.P. is 0.6, 1.7. 2.8.

Here a = 0.6, d = 1.7 – 0.6 = 1.1 and n = 100

S_{n} = \(\frac{n}{2}\) [2a+(n – 1) d]

∴ S_{100} = \(\frac{100}{2}\) [2(0.6) + (100 – 1) 1.1]

= 50 [1.2 + 108.9] = 5505.

(iv) Given AP. is \(\frac{1}{5}\), \(\frac{1}{12}\), \(\frac{1}{10}\), ………… to 11 terms.

Here a = \(\frac{1}{5}\), d = \(\frac{1}{5}\) and n = 11

S_{n} = \(\frac{n}{2}\) [2a +(n – 1)d]

∴ S_{11} = \(\frac{11}{2}\left[2\left(\frac{1}{15}\right)+(11-1) \frac{1}{60}\right]\)

= \(\frac{11}{2}\left[\frac{2}{15}+\frac{10}{60}\right]=\frac{11}{2}\left[\frac{2}{15}+\frac{1}{6}\right]\)

= \(\frac{11}{2}\left[\frac{4+5}{30}=\frac{9}{30}\right]=\frac{33}{20}\).

Question 2.

Find the sums given below:

(i) 7+ 10\(\frac{1}{2}\) + 14 + …………… + 84

(ii) 34 + 32 + 30 + ………. + 10

(iii) – 5 + (- 8) + (- 11) +………+ (- 230)

Solution:

(i) Given A.P. is

7 + 10\(\frac{1}{2}\) + 14 + ………….. + 84

Here a = 7, d = 10\(\frac{1}{2}\) – 7 = \(\frac{21}{2}\) – 7

= \(\frac{21-14}{2}=\frac{7}{2}\)

and l = T_{n} = 84

or a + (n – 1) d = 84

or 7 + (n – 1) \(\frac{7}{2}\) = 84

or (n – 1) \(\frac{7}{2}\) = 84 – 7 = 77

or n – 1 = 77 × \(\frac{2}{7}\) =22

n = 22 + 1 = 23

∵ S_{n} = \(\frac{n}{2}\) [a + l]

Now, S_{23} = \(\frac{23}{2}\) [7 + 84]

= \(\frac{23}{2}\) × 91 = \(\frac{2093}{2}\).

(ii) Given A.P. is 34 + 32 + 30 + …………… + 10

Here a = 34, d = 32 – 34 = -2

l = T_{n} = 10

a + (n – 1) d = 10

or 34 + (n – 1) (- 2) = 10

or – 2(n – 1) = 10 – 34 = -24

or n – 1 = 12

n = 12 + 1 = 13

[∵ S_{n} = \(\frac{n}{2}\) [a + l]]

Now, S_{13} = \(\frac{13}{2}\) [34 + 10]

= \(\frac{23}{2}\) × 44

= 13 × 22 = 286.

(iii) Give A.P. is – 5 + (- 8) + (- 11) + ……………. + (- 230)

Here a = – 5, d = – 8 + 5 = – 3

and l = T_{n} = – 230

a + (n – 1) d = – 230

or – 5 + (n – 1) (- 3) = – 230

or – 3(n – 1) = – 230 + 5 = – 225

or n – 1 = \(\frac{225}{3}\) = 75

or n = 75 + 1 = 76

Now, S_{76} = \(\frac{76}{2}\) [- 5 + (- 23o)]

= 38 (- 235) = – 8930.

Question 3.

In an AP:

(i) given a = 5, d = 3, a_{n} = 50 find n and S_{n}.

(ii) given a = 7. a_{13} = 35, find d and S_{13}.

(iii) given a_{12} = 37, d = 3. find a and S_{12}.

(iv) given a_{3} = 15, S_{10} = 125, find d and a_{10}.

(y) given d = 5, S_{9} = 75, find a and a_{9}.

(vi) given a = 2, d = 8, S_{n} = 90, find n and a_{n}.

(vii) given a = 8, a_{n} = 62, S_{n} = 210, find n and d.

(viii)given a_{n} = 4, d = 2, S_{n} = -14, find n and a.

(ix) given a = 3, n = 8, S = 192, find d.

(x) given l = 28, S = 144, and there are total 9 terms. Find a.

Solution:

(i) Given a = 5, d = 3, a_{n} = 50

a_{n} = 50

a + (n – 1) d = 50

or 5 + (n – 1) 3 = 50

or 3 (n – 1) = 50 – 5 = 45

or n – 1 = \(\frac{45}{3}\) = 15

or n = 15 + 1 = 16

Now, S_{n} = latex]\frac{n}{2}[/latex] [a + l]

= latex]\frac{16}{2}[/latex] [5 + 50] = 8 × 55 = 440.

(ii) Given a = 7, a_{13} = 35

a_{13} = 35

a + (n – 1) d = 35

or 7 + (13 – 1) d = 35

or 12 d = 35 – 7 = 28

or d = \(\frac{7}{3}\).

∵ [S_{n} = latex]\frac{n}{2}[/latex] [a + l]

Now, S_{13} = \(\frac{13}{2}\) [7 + 35]

= \(\frac{13}{2}\) × 42 = 273

(iii) Given a_{12} = 37, d = 3

∵ a_{12} = 37

a + (n – 1) d = 37

or a + (12 – 1) 3 = 37

a + 33 = 37

a = 37 – 33 = 4

∵ [S_{n} = latex]\frac{n}{2}[/latex] [a + l]

Now, S_{12} = \(\frac{13}{2}\) [4 + 37]

= 6 × 41 = 246.

(iv)Given a_{3} = 15, S_{10} = 125

∵ a_{3} = 15

a + (n – 1) d = 15

a + (3 – 1) d = 15

or a + 2d = 15 …………….(1)

∵ S_{10} = 125

∵ [S_{n} = latex]\frac{n}{2}[/latex] [2a + (n – 1)d]

\(\frac{10}{2}\) [2a + (10 – 1) d] = 125

or 5[2a + 9d] = 125

Note this

or 2a + 9d = \(\frac{125}{5}\) = 25

or 2a + 9d = 25 …………(2)

From (1), a = 15 – 24 …………..(3)

Substitute this value of (a) in (2i, we c1

2(15 – 2d) + 9d = 25

or 30 – 4d + 9d = 25

5d = 25 – 30

or d = \(\frac{-5}{5}\) = -1

Substitute this value of d in (3), we get

a = 15 – 2(- 1)

a = 15 + 2 = 17

Now, a_{10} = 17 + (10 – 1)(- 1)

∵ T_{n} = a + (n – 1) d = 17 – 9 = 8.

(v) Given d = 5, S_{9} = 75

∵ S_{9} = 75

∵ [S_{n} = \(\frac{n}{2}\) [2a + (n – 1)d]

\(\frac{9}{2}\) [2a + 40] = 75

or [2a + 40] = \(\frac{50}{3}\)

or 2a = \(\frac{50}{3}\) – 40

or a = \(-\frac{70}{3} \times \frac{1}{2}\)

or a = – \(\frac{35}{3}\)

Now, a_{9} = a + (n – 1) d

= – \(\frac{35}{3}\) + (9 – 1)^{0} × 5

= – \(\frac{35}{3}\) + 4o = \(\frac{-35+120}{3}\)

a_{9} = \(\frac{85}{3}\).

(vi) Given a = 2, d = 8, S_{n} = 90

∵ S_{n} = 90

\(\frac{n}{2}\) [2a + (n – 1)d] = 9o

or \(\frac{n}{2}\) [2 × 2 + (n – 1) 8] = 90

or n [2 + 4n – 4] = 90

or n (4n – 2) = 90

or 4n^{2} – 2n – 90 = 0

or 2n^{2} – 10n + 9n – 45 = 0

S = – 2

P = – 45 × 2 = – 90

or 2n [n – 5] + 9(n – 5) = 0

or (2n + 9) (n – 5) = 0

Either 2n + 9 = 0 or n – 5 = 0

Either n = – \(\frac{9}{2}\) or n = 5

∵ n cannot be negative so reject n = – \(\frac{9}{2}\)

∴ n = 5

Now, a_{n} = a_{5} = a + (n – 1) d

= 2 + (5 – 1) 8 = 2 + 32 = 34.

(vii) Given a = 8, a_{n} = 62, S_{n} = 210

∵ S_{n} = 210

\(\frac{n}{2}\) [a + a_{n}] = 210

or \(\frac{n}{2}\) [8 + 62] = 210

or \(\frac{n}{2}\) × 70 = 210

or n = \(\frac{210}{35}\) = 6

Now, a_{n} = 62

[∵ T_{n} = a + (n – 1) d]

or 5d = 62 – 8 = 54

or d = \(\frac{54}{5}\).

(viii) Given a_{n} = 4, d = 2, S_{n} = – 14

∵ a_{n} = 4

a + (n – 1) d = 4

or a + (n – 1)2 = 4

or a + 2n – 2 = 4

or a = 6 – 2n ……………(1)

and S_{n} = – 14

\(\frac{n}{2}\) [a + a_{n}] = – 14

or \(\frac{n}{2}\) [6 – 2n +4] = – 14 (Using (1))

or \(\frac{n}{2}\) [10 – 2n] = – 14

or 5n – n^{2} + 14 = 0

or n^{2} – 5n – 14 = 0

S = – 5

P = 1 × – 14 = – 14

or n^{2} – 7n + 2n – 14 = 0

or n(n – 7) + 2 (n – 7) = 0

or (n – 7) (n + 2) = 0

either n – 7 = 0

or n + 2 = 0

n = 7 or n = -2

∵ n cannot be negative so reject n = – 2

∴ n = 7

Substitute this value of n in (1), we get

a = 6 – 2 × 7

a = 6 – 14 = – 8.

(ix) Given a = 3, n = 8, S = 192

∵ S = 192

S_{8} = 192 [∵ n = 8]

∵ S_{n} = \(\frac{n}{2}\) [2a + (n – 1)d]

or \(\frac{8}{2}\) [2 × 3 + (8 – 1) d] = 192

or 4 [6 + 7d] = 192

6 + 7d = \(\frac{192}{4}\) = 48

or 7d = 48 – 6 = 42

or d = \(\frac{42}{6}\) = 6.

(x) Given l = 28, S = 144 and there are total 9 terms

∴ n = 9; l = a_{9} = 28; S_{9} = 144

∵ a_{9} = 28

or a + (9 – 1) d = 28

∵ [a_{n} = T_{n} = a + (n – 1) d ]

or a + 8d = 28 …………….(1)

and S_{9} = 144

∵ [S_{n} = \(\frac{n}{2}\) [a + a_{n}]]

\(\frac{9}{2}\) [a + 28] = 144

or a + 28 = \(\frac{144 \times 2}{9}\) = 32

a = 32 – 28 = 4.

Question 4.

How many terms of the A.P : 9, 17, 5… must be taken to give a sum of 636?

Solution:

Given A.P. is 9, 17, 25, …………

Here a = 9, d = 17 – 9 = 8

But S_{n} = 636

\(\frac{n}{2}\) [2a + (n – 1) d] = 636

or \(\frac{n}{2}\) [2(9) + (n – 1) 8] = 636

or \(\frac{n}{2}\) [18 + 8n – 8] = 636

or n [4n + 5] = 636

or 4n^{2} + 5n – 636 = 0

a = 4, b = 5, c = – 636

D = (5) – 4 × 4 × (- 636)

= 25 + 10176 = 10201

∴ n = \(\frac{-b \pm \sqrt{\mathrm{D}}}{2 a}\)

= \(\frac{-5 \pm \sqrt{10201}}{2 \times 4}=\frac{-5 \pm 101}{8}\)

= \(\frac{-106}{8} \text { or } \frac{96}{8}\)

= \(-\frac{53}{4}\) or 12

∴ n cannot be negative so reject n = \(-\frac{53}{4}\)

∴ n = 12

Hence, sum of 12 terms of given A.P. has sum 636.

Question 5.

The first term ofan AP is 5, the last term is 45 and the sum Is 400. Find the number of terms and the common difference.

Solution:

Given that a = T_{1} = 5; l = a_{n} = 45

and S_{n} = 400

∴ T_{n} = 45

a + (n – 1) d = 45

or 5 + (n – 1) d = 45

or (n – 1) d = 45 – 5 = 40

or (n – 1) d = 40 ……………….(1)

and S_{n} = 400

\(\frac{n}{2}\) [a + a_{n}] = 400

or \(\frac{n}{2}\) [5 + 45] = 400

or 25 n = 400

or n = \(\frac{400}{25}\) = 16

Substitute this value of n in (1), we get

(16 – 1) d = 40

or 15d = 40

d = \(\frac{40}{15}\) = \(\frac{8}{3}\)

Hence, n = 16 and d = \(\frac{8}{3}\)

Question 6.

The first and last terms of an AP are 17 and 350 respectively. 1f the common difference is 9, how many terms are there

and what is their sum?

Solution:

Given that a = T_{1} = 17;

l = a_{n} = 350 and d = 9

∵ l = a_{n} = 350

a + (n – 1) d = 350

17 + (n – 1) 9 = 350

or 9 (n – 1) = 350 – 17 = 333

or n – 1 = \(\frac{333}{9}\) = 37

n = 37 + 1 = 38

Now, S_{38} = \(\frac{n}{2}\) [a + l]

= \(\frac{38}{2}\) (17 + 350]

= 19 × 367 = 6973.

Hence, sum of 38 terms of given A.P. arc 6973.

Question 7.

Find the sum of first 22 terms of an AP. in which d = 7 and 22nd term is 149.

Solution:

Given that d = 7; T_{22} = 149 and n = 22

∵ T_{22} = 149

a + (n – 1) d = 149

or a + (22 – 1) 7 = 149

or a + 147 = 149

or a = 149 – 147 = 2

Now, S_{22} = [a + T_{22}]

= \(\frac{22}{2}\) [2+ 149] = 11 × 151 = 1661

Hence, sum of first 22 temis of given A.P. is 1661.

Question 8.

Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

Solution:

Let ‘a’ and ‘d’ be fïrst term and common difference

Given that T_{2} = 14; T_{3} = 18 and n = 51

∵ T_{2} = 14

a + (n – 1) d = 14

a + (2 – 1)d = 14

or a + d = 14

a = 14 – d …………….(1)

and T_{3} = 18 (Given)

a + (n – 1) d = 18

a + (3 – 1) d = 18

or a + 2d = 18

or 14 – d + 2d = 18

or d = 18 – 14 = 4

or d = 4

Substitute this value of d in (1), we get

a = 14 – 4 = 10

Now, S_{51} = \(\frac{n}{2}\) [2a + (n – 1) d]

= \(\frac{51}{2}\) [2 × 10 + (51 – 1) 4]

= \(\frac{51}{2}\) [2o + 2oo]

= \(\frac{51}{2}\) × 220 = 51 × 110 = 5610

Hence, sum of first 51 terms of given A.P. is 5610.

Question 9.

If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Solution:

Let ‘a’ and ‘d’ be the first term and common difference of given A.P.

According to 1st condition

S_{7} = 49

\(\frac{n}{2}\) [2a + (n – 1) d] = 49

or \(\frac{7}{2}\) [2a + (7 – 1) d] = 49

or \(\frac{7}{2}\) [2a + 6d] = 49

or a + 3d = 7

or a = 7 – 3d

According to 2nd condition

S_{17} = 289

\(\frac{n}{2}\) [2a+(n – 1)d]=289

\(\frac{17}{2}\) [2a + (17 – 1) d] = 289

a + 8d = \(\frac{289}{17}\) = 17

Substitute the value of a from (1), we get

7 – 3d + 8d = 17

5d = 17 – 7 = 10

d = \(\frac{10}{5}\) = 2

Substitute this value of d in (1), we get

a = 7 – 3 × 2

a = 7 – 6 = 1

Now, S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

= \(\frac{n}{2}\) [2 × 1 + (n – 1) × 2] = n [I + n – 1] = n × n = n^{2}

Hence, sum of first n terms of given A.P. is n^{2}.

Question 10.

Show that a_{1}, a_{2}, ……., a_{n}, … form an AP where is defined as below.

(i) a_{n} = 3 + 4n

(ii) a_{n} = 9 – 5n

Also find the sum of the first 15 terms In each case.

Solution:

(i) Given that a_{n} = 3 + 4n ………..( 1)

Putting the different values of n in(1), we get

a_{1} = 3 + 4(1) = 7;

a_{2} = 3 + 4 (2) = 11

a_{3} = 3 + 4 (3) = 15, …………

Now, a_{2} – a_{1} = 11 – 7

and a_{3} – a_{2} = 15 – 11 = 4

a_{2} – a_{1} = 11 – 7 = 4

and a_{3} – a_{2} = 4 = d(say)

∵ given sequence form an A.P.

Here a = 7, d = 4 and n = 15

∴ S_{15} = \(\frac{n}{2}\) [2a + (n – 1)d]

= \(\frac{15}{2}\) [2(7) + (15 – 1) 4]

= \(\frac{15}{2}\) [14 + 56] = \(\frac{15}{2}\) × 70

= 15 × 35 = 525.

(ii) Given that a_{n} = 9 – 5n ……………….(1)

Putting the different values of n is (1), we get

a_{1} = 9 – 5(1) = 4;

a_{2} = 9 – 5(2) = -1;

a_{3} = 9 – 5(3) = -6.

Now, a_{2} – a_{1} = – 1 – 4 = – 5

and a_{3} – a_{2} = – 6 + 1 = – 5

∵ a – a_{1} = a_{3} – a_{2} = – 5 = d (say)

∴ given sequence form an A.P.

Here a = 4, d = – 5 and n = 15

∴ S_{15} = \(\frac{n}{2}\) [2a + (n – 1) d]

= \(\frac{15}{2}\) [2(4) + (15 – 1) (-5)]

= \(\frac{15}{2}\) [8 – 70]

= \(\frac{15}{2}\) (-62) = – 465

Question 11.

If the sum of the first n terms of an AP is 4n – n^{2}, what is the first term (that is S_{1}) ? What is the sum of two terms? What is the second term ? Similarly, find the 3rd, the 10th and the nth terms.

Solution:

Given that, sum of n terms of an A.P. are

S_{n} = 4n – n^{2}

Putting n = 1 in (1), we get

S_{1} = 4(1) – (1)^{2} = 4 – 1

S_{1} = 3

∴ a = T_{1} = S_{1} = 3

Putting n = 2, in (1), we get

S_{2} = 4(2) – (2)^{2} = 8 – 4

S_{2} = 4

or T_{1} + T_{2} = 4

or 3 + T_{2} = 4

or T_{2} = 4 – 3 = 1

Putting n = 3 in (1), we get

S_{3} =4(3) – (3)^{2} = 12 – 9

S_{3} = 3

or S_{2} + T_{3} = 3

or 4+ T_{3} = 3

or T_{3} = 3 – 4 = – 1

Now, d = T_{2} – T_{1}

d = 1 – 3 = -2

∴ T_{10} = a + (n – 1) d

= 3 + (10 – 1) (- 2)

T_{10} = 3 – 18 = – 15

and T_{n} = a + (n – 1) d

= 3 + (n – 1) (- 2)

= 3 – 2n + 2

T_{n} = 5 – 2n.

Question 12.

Find the sum of the first 40 positive integers divisible by 6.

Solution:

Positive integers divisible by 6 are 6, 12, 18, 24, 30, 36 42, …

Here a = T_{1} = 6, T_{2} = 12, T_{3} = 18, T_{4} = 24

T_{2} – T_{1} = 12 – 6 = 6

T_{3} – T_{2} = 18 – 12 = 6

T_{4} – T_{3} = 24 – 18 = 6

∵ T_{2} – T_{1} = T_{3} – T_{2} = T_{4} – T_{3} = 6 = d (say)

Using formula, S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

S_{40} = \(\frac{40}{2}\) [2(6) + (40 – 1)6].

= 20 [12 + 234]

= 20 (246) = 4920

Hence, sum of first 40 positive integers divisible by 6 is 4920.

Question 13.

Find the sum of first 15 multiples of 8.

Solution:

Multiples of 8 are 8, 16, 24, 32, 40, 48, …………..

Here a = T_{1} = 8; T_{2} = 16; T_{3} = 24 ; T_{4} = 32

T_{2} – T_{1} = 16 – 8= 8

T_{3} – T_{2} = 24 – 16=8

T_{2} – T_{1} = T_{3} – T_{2} = 8 = d(say)

Ùsing formula. S_{n} = [2a + (n – 1) d}

S_{15} = \(\frac{15}{2}\) [2(8) + (15 – 1) 8]

= \(\frac{15}{2}\) [16 + 112]

= \(\frac{15}{2}\) × 128 = 960

Hence, sum of first 15 multiples of 8 is 960.

Question 14.

Find the sum of the odd numbers between 0 and 50.

Solution:

Odd numbers between 0 and 50 are 1, 3, 5, 7, 9, …………, 49

Here a = T_{1} = 1; T_{2} = 3; T_{3} = 5 ; T_{4} = 7

and l = T_{n} = 49

T_{2} – T_{1} = 3 – 1 = 2

T_{3} – T_{2} = 5 – 3 = 2

∵ T_{2} – T_{1} = T_{3} – T_{2} = 2 = d (say)

Also, l = T_{n} = 49

a + (n – 1) d = 49

1 + (n – 1) 2 = 49

or 2 (n – 1) = 49 – 1 = 48

n – 1 = \(\frac{48}{2}\) = 24

n = 24 + 1 = 25.

Using formula, S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

S_{25} = \(\frac{25}{2}\) [2(1) + (25 – 1)2]

= \(\frac{25}{2}\) [2 + 48]

= \(\frac{25}{2}\) × 50 = 625

Hence, sum of the odd numbers between 0 and 50 are 625.

Question 15.

A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ 200 for

the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc., the penalty for each succeeding day being ₹ 50 more than for the preceding day, How much money the contractor has to pay as penalty, if he hasdelayed the work by 30 days?

Solution:

Penalty (cost) for delay of one, two, third day are ₹ 200, ₹ 250, ₹ 300

Now, penalty increase with next day with a difference of ₹ 50.

∴ Required A.P. are ₹ 200, ₹ 250, ₹ 300, ₹ 350, …

Here a = T_{1} = 200; d = ₹ 50 and n = 30

Amount of penalty gives after 30 days

= S_{30}

= \(\frac{n}{2}\) [2a + (n – 1) d]

= \(\frac{30}{2}\) [2(2oo) + (30 – 1) 50)

= 15 [400 + 1450]

= 15 (1850) = 27750

Hence, ₹ 27350 pay as penalty by the contractor if he has delayed the work 30 days.

Question 16.

A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If

each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.

Solution:

Let amount of prize given to 1st student = ₹ x

Amount of prize given to 2nd student = ₹ (x – 20)

Amount of prize given to 3rd student = ₹ [x – 20 – 20] = ₹ (x – 40)

and so on.

∴ Required sequence are ₹ x, ₹ (x – 20), ₹ (x – 40), … which form on A.P. with

a = ₹ x, d = – 20 and n = 7

Using formula, S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

S_{7} = \(\frac{7}{2}\) [2(x) + (7 – 1) (- 20)]

S_{7} = \(\frac{7}{2}\) [2x – 120] = 7 (x – 60)

According to question,

7 (x – 60) = 700

x – 60 = 7 = 100

x – 60 = 7 = 100

x = 100 + 60

x = 160

Hence, 7 prizes are ₹ 160, ₹ 140, ₹ 120, ₹ 100, ₹ 80, ₹ 60, ₹ 40.

Question 17.

In a school, student thought of planting trees in and around the school to reduce air pollution. It was decided that

number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g. – a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

Solution:

Number of trees planted by three sections of class I = 3 × 1 = 3

Number of trees planted by three sections of class II = 3 × 2 = 6

Number of trees planted by three sections of class III = 3 × 3 = 9

…………………………………………………………………..

…………………………………………………………………..

……………………………………………………………………

Number of trees planted by three sections of class XII = 3 × 12 = 36

:. Required A.P. are 3, 6, 9, …………., 36

Here a = T_{1} = 3; T_{2} = 6; T_{3} = 9

and l = T_{n} = 36; n = 12

d = T_{2} – T_{1} = 6 – 3 = 3

Total number of trees planted by students

= S_{12}

= \(\frac{n}{2}\) [a + l]

= \(\frac{12}{2}\) [3+ 36]

= 6 × 39 = 234

Hence, 234 trees will be planted by students to reduce air pollution.

Question 18.

A spiral is made up of successive semicircics, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, …. as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircies? (Take π = \(\frac{22}{7}\))

[Hint: Length of successive semicircies is ‘l_{1}’ ‘l_{2}’ ‘l_{3}‘ ‘l_{4}‘ … wIth centres at A, B, A, B, …, respectively.]

Solution:

Let l_{1} = length of first semi circle = πr_{1} = π(0.5) = \(\frac{\pi}{2}\)

l_{2} = length of second semi circlem = πr_{2}= π(1) = π

l_{3} = length of third semi circle = πr_{3} = π(1.5) = \(\frac{3 \pi}{2}\)

and l_{4} = length of fourth senil circle = πr_{4} = π(2) = 2π and so on

∵ length of each successive semicircle form an A.P.

Here

a = T_{1} = \(\frac{\pi}{2}\); T_{2} = π;

T_{3} = \(\frac{3 \pi}{2}\); T_{4} = 2π……… and n = 13

d = T_{2} – T_{1} = π – \(\frac{\pi}{2}\)

= \(\frac{2 \pi-\pi}{2}=\frac{\pi}{2}\)

Length of whole spiral = S_{13}

Hence, total length of a spiral made up of thirteen consecutive semi circies is 143 cm

Question 19.

200 logs are stacked in the following manner : 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see Fig). In how many rows are the 200 logs placed and how many logs are in the top row?

Solution:

Number of logs in the bottom (1st row) = 20

Number of logs in the 2nd row = 19

Number of log in the 3rd row = 18 and so on.

∴ Number of logs in the each steps form an

Here a = T_{1} = 20; T_{2} = 19; T_{3} = 18…

d = T_{2} – T_{1} = 19 – 20 = – 1

Let S, denotes the number logs.

Using formula, S_{n} = \(\frac{n}{2}\) [2a + (n – 1) d]

= \(\frac{n}{2}\) [2 (20) + (n – 1) (-1)]

∴ S_{n} = \(\frac{n}{2}\) [40 – n + 1]

= \(\frac{n}{2}\) [41 – n]

According to question,

\(\frac{n}{2}\) [41 – n] = 200

or 41 – n^{2} = 400

or – n^{2} + 41n – 400 = 0

or n^{2} – 41n + 400 = 0

S = – 41, P = 400

or n^{2} – 16n – 25n + 400 = 0

or n (n – 16) – 25 (n – 16)=0

or (n – 16) (n – 25) = 0

Either n – 16 = 0 or n – 25 = 0

Either n = 16 or n = 25

∴ n = 16, 25.

Case I:

When n = 25

T_{25} = a + (n – 1) d

= 20 + (25 – 1)(- 1)

= 20 – 24 = – 4;

which is impossible

∴ n = 25 rejected.

Case II. When n = 16

T_{16} = a + (n – 1) d

= 20 + (16 – 1)(- 1)

= 20 – 15 = 5

Hence, there are 16 row and 5 logs are in the top row.

Question 20.

In a potato race a bucket ¡s placed at the starting point which Ls 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line (see Fig.)

Each competitor starts from the bucket, picks up the nearest potato, runs back with It, drops It in the bucket, runs back to pick up the next potato, runs to the bucket to drop it In, and the continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

(Hint : To pick up the first potato and second potato, the distance run is [2 × 5 + 2 × (5 + 3)] m)

Solution:

Distance covered to pick up the Ist potato = 2(5) m = 10 m

Distance between successive potato = 3 m

distance covered to pick up the 2nd potato = 2(5 + 3) m = 16 m

Distance covered to pick up the 3rd potato = 2 (5 + 3 + 3) m = 22 m

and this process go on. h is clear that this situation becomes an A.P. as 10 m, 16 m, 22 m, 28 m, …

Here a = T_{1} = 10; T_{2} = 16; T_{3} = 22, …

d = T_{2} – T_{1} = 16 – 10 = 6 and n = 10

∴ Total distance the competitor has to run = S_{10}

= \(\frac{n}{2}\) [2a + (n – 1) d]

= \(\frac{10}{2}\) [2(10) + (10 – 1) 6]

= 5 [20 + 54]

= 5 × 74 = 370

Hence, 370 m is the total distance run by a competitor.