KSEEB Solutions for Class 10 Maths Chapter 1 Arithmetic Progressions Ex 1.3

Students can Download Class 10 Maths Chapter 1 Arithmetic Progressions Ex 1.3 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 10 Maths helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

Karnataka State Syllabus Class 10 Maths Chapter 1 Arithmetic Progressions Ex Ex 1.3

Question 1.
FInd the sum of the following APs:
i) 2, 7, 12, ……… to 10 terms
ii) -37, -33, -29,………. to 12 terms.
iii) 0.8, 1.7, 2.8 to 100 terms.
iv)  terms
Answer:
i) Sum of 2, 7, 12, ………. 10?
a = 2, d = 7 – 2 = 5, n = 10
\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
\(s_{10}=\frac{10}{2}[2 \times 2+(10-1) 5]\)
= 5[4 + 9 × 5]
= 5[4 + 45]
= 5 × 49
∴ S₁₀ = 245

ii) -37, -33, -29, to 12……. terms.
Solution:
a = -37, d = -33 – (-37) = -33 + 37
n = 12, d = 4
\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
\(S_{12}=\frac{12}{2}[2 \times -37+(12-1) 4]\)
= 6[-74 + 11 × 4]
= 6[-74 + 44]
= 6 × -30
∴ S₁₂ = -180.

iii) 0.6, 1.7, 2.8,……… , to 100 terms.
a = 0.6, d = 1.7 – 0.6 = 1.1
n = 100, S₁₀₀ =?
\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
\(S_{100}=\frac{100}{2}[2 \times 0.6+(100-1)(1.1)]\)
= 50[1.2 + 99(1.1)]
= 50[1.2 + 108.9]
= 50 × 110.1
∴ S₁₀₀ = 5505
Answer:

Question 2.
Find the sums given below:
i) 
ii) 34 + 32 + 30 + ……….+ 10
iii) -5 + (-8) + (-11) + ……….. +(-230)
Answer:

ii) 34 + 32 + 30 + ………. + 10
Solution:
a = 34, d = 32 – 34 =-2, a_n = 10,
S_n =?
a + (n – 1)d = a_n
34 + (n – 1)(-2) = 10
34 – 2n + 2 = 10
-2n + 36 = 10
-2n = 10 – 36
-2n = -26
2n = 26
\(n=\frac{26}{2}\)
∴ n = 13
\(S_{n}=\frac{n}{2}[a+l]\)
\(\mathrm{S}_{23}=\frac{13}{2}[34+10]\)
\(=\frac{13}{2} \times 44\)
= 13 × 22
∴ S₂₃ = 286.

iii) -5 + (-8) + (-11) +……… + (-230)
Solution:
a = -5, d = -8 – (-5) = -8 + 5
a, = -230.
S_n =? d = -3
a+(n – 1)d = a_n
-5 + (n – 1)(-3) = -230
-5 – 3n + 3 = -230
-3n – 2 = -230
-3n = -230 +2
-3n = -228
3n = 228
∴ n = 228/3
∴ n = 76.
\(\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[\mathrm{a}+\mathrm{a}_{\mathrm{n}}\right]\)
\(S_{ 76 }=\frac { 76 }{ 2 } \left[ -5-230 \right]  \)
\(=\frac{76}{2} \times -235\)
= 38 × -235
∴ S₇₆= -8930.

Question 3.
In an AP:
i) given a = 5. d = 3. a_n = 50, find ‘n’ and S_n.
ii) given a = 7, a₁₃ = 35, find d’ and S₁₃.
iii) given a₁₂= 37, d = 3. find ‘a’ and S₁₂.
iv) given a₃ = 15, S₁₀ = 125. find ‘d’ and a₁₀.
v) given d = 5. S₉ = 72, fInd ‘a’ and a₉.
vi) given a = 2, d = 8, S_n = 90, fInd ‘n and a₁₁.
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’.
Answer:
i) a = 5, d = 3, a_n = 50. n=?, S_n =?
a + (n – 1)d = a_n
5 + (n – 1) 3 = 50
5 + 3n – 3 = 50
3n + 2 = 50
3n = +50 – 2
3n = 48
\(n=\frac{48}{3}\)
∴ n = 16
\(S_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(S_{16}=\frac{16}{2}[5+50]\)
\(=\frac{16}{2}[55]\)
= 8 × 55
∴ S₁₆= 440.

ii) a = 7, a₁₃ = 35. d =?. S₁₃ =?
Solution:
a+(n – 1) d = a_n
7 + (13 – 1) d = 35
7 + 12d = 35
12d = 35 – 7
12d = 28
\(d=\frac{28}{12}\)
\(d=\frac{7}{3}\)
\(S_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(S_{n}=\frac{7}{3} \times \frac{1}{2}[7+35]\)
\(\mathrm{s}_{\mathrm{n}}=\frac{7}{6} \mathrm{I} 42 \mathrm{l}\)
= 7 × 7
∴ S_n = 49

iii) a₁₂= 37, d = 3, a =? S₁₂ =?
a + (n – 1)d = a_n
a + (12 -1) 3 = 37
a + 11 × 3 = 37
a + 33 = 37
∴ a = 37 – 33
∴ a = 4
\(S_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(\mathrm{S}_{12}=\frac{12}{2}[4+37]\)
= 6[4 + 37]
∴ S₂ = 246

iv) a₃ = 15, S₁₀ = 125. d =?, a₁₀=?
Solution:
a₃ = a + 2d = 15
∴ a= 15 – 2d
\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
\(s_{10}=\frac{10}{2}[2(15-2 d) \div(10-1) d]=125\)
= 5[30 – 4d + 9d] = 125
= 5[30 + 5d] = 125
150 + 25d = 125
25d = 125 + 150
25d = -25
\(\mathrm{d}=\frac{-25}{25}\)
∴ d = -1.
Substittuting the value of ‘d’
a = 15 – 2d
= 15 – 2(-1)
= 15 + 2
∴ a = 17.
∴ a₁₀ = a + 9d
= 17 + 9(-1)
= 17 – 9
∴ a₁₀ = 8

v) d = 5, S₉ = 72. a =? a₉ =?
Solution:
\(\mathrm{s}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}]\)
\(s_{9}=\frac{9}{2}[2 \times a+(9-1) 5]=72\)
\(\frac{9}{2}[2 a+8 \times 5]=72\)
\(\frac{9}{2}[2 a+40]=72\)
\(2 a+40=72 \times \frac{2}{9}\)
2a + 40 = 16
2a = 16 – 40
2a = -24
\(a=\frac{-24}{2}\)
∴ a = -12
a₉ = a + (n – 1) d
= -12 + (9 – 1) (5)
= -12 + 8 × 5
= -12 + 40
∴ a₉ = 28

vi) a = 2, d = 8 S_n = 90, n =?, a₃ =?
Answer:



n (8n – 4) = 90 × 2/1
8n² – 4n = 180
8n² – 4n = 1800
2n² – n – 45 = 0

2n² – 10n + 9n – 45 = 0
2n(n – 5) + 9(n – 5) = 0
(n – 5)(2n + 9) = 0
If n – 5 = 0 then n = 5
∴ a_n = a + (n – 1) d
a₅= 2 + (5 – 1) 8
= 2 + 4 × 8
= 2 + 32
∴ a₅ = 34

vii) a = 8, a_n = 62, S_n = 210, n =?, d =?
Answer:




∴ n = 6
a_n = 62
a₆= a + 5d = 62
8 + 5d = 62
5d = 62 – 8
5d = 54

viii) a = 4, d = 2, S_n = -14, n =?, a =?
Answer:
a_n = a + (n – 1) d = 4
a + (n – 1) 2 = 4
a + 2n – 2 = 4
a + 2n = 4 + 2
a = 2n = 6 …………. (i)



– 2n² + 10n = -28
– 2n² + 10n + 28 = 0
2n² – 10n – 28 = 0 – 14
²ⁿ² – 5n – 14 = 0

n² – 7n + 2n – 14 = 0
n(n – 7) + 2 (n – 7) = 0
(n – 7) (n + 2) = 0
If n – 7 = 0 then. n = 7
a = – 2n + 6
= – 2 × 7 + 6
= – 14 + 6
∴ a = -8
∴ n = 7, a = – 8.

ix) a = 3, n = 8, S = 192, d =?
\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
\(S_{8}=\frac{8}{2}[2 \times 3+(8-1) d]=192\)
\(\frac { 8 }{ 2 } [6+7\times d]=192\)
4 (6 + 7d) = 192
\(6+7 d=\frac{192}{4}\)
6 + 7d = 48
7d = 48 – 6
7d = 42
\(d=\frac{42}{7}\)
∴ d = 6

x) l = a_n = 28, S = 144, n = 9, a =?
Solution:
\(s_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(s_{9}=\frac{9}{2}[a+28]=144\)
\(\frac{9}{2}[a+28]=144\)
\(a+28=144 \times \frac{2}{9}\)
a + 28 = 32
a = 32 – 28
∴ a = 4.

Question 4.
How many terms of the AP: 9, 17, 25, … must be taken to give a sum of 636?
Answer:
9 + 17 + 25 + ……. + a_n = 636
a = 9, d = 17 – 9 = 8, S_n= 636, n =?




n(8n + 10) = 636 × 2
8n² + 10n = 1272
8n²+ 10n – 1272 = 0
4n² + 5n – 636 = o
4n²+ 5n – 48n – 636 = 0

n(4n + 5) – 12(4n + 5) = 0
(4n + 5) (n – 12) = 0
if n – 12 = 0 then, n = 12
∴ n = 12.

Question 5.
The first term of an AP is 5. the last term is 45 and the sum is 400. Find the number of terms and the common difference.
Answer:
a = 5, a_n = 45, S_n= 400, n =?, d =?
\(S_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(=\frac{n}{2}[5+45]=400\)
\(\frac{n}{2} \times 50=400\)
n × 50 = 800
\(n=800 \times \frac{1}{50}\)
∴ n = 16
a_n = a + (n – 1) d
a₁₆= 5 + (16 – 1)d = 45
5 + 15d = 45
15d = 45 – 5
15d = 40
\(d=\frac{40}{15}=\frac{8}{3}\)
\(\mathrm{n}=16, \mathrm{d}=\frac{8}{3}\)

Question 6.
The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
Answer:
a = 17, a_n= 350, d = 9, n =?, S_n =?
a_n = a + (n – 1) d
17 + (n – 1)9 = 350
17 + 9n – 9 = 350
9n + 8 = 350
9n = 350 – 8
9n = 342
\(\quad n=\frac{342}{9}\)
∴ n = 38
\(S_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(S_{38}=\frac{38}{2}[17+350]\)
= 19 × 367
∴ S₃₈ = 6973
∴ n = 38, S₃₈ = 6973

Question 7.
Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
Answer:
d = 7, a₂₂ = 149, S₂₂ =?
a = a + (n – 1)d
a + (22 – 1)7 = 149
a + 21 × 7 = 149
a + 147 = 149
=149 – 147
∴ a = 2
\(S_{n}=\frac{n}{2}\left[a+a_{n}\right]\)
\(S_{22}=\frac{22}{2}[2+149]\)
= 11[151]
∴ S₂₂ = 1661

Question 8.
Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
Answer:
If a₂ = 14, a₃ = 18, then S₁₁ =?
a₂ = 14
a + d = 14 …………. (i)
a₃ = 18
a + 2d = 18 …………… (ii)
from equation (1) – equatIon (11),

d =4.
Substituting the value of ‘d’ In eqn. (i)
a + d = 14
a + 4 = 14
a = 14 – 4
∴ a = 10
a = 10, d = 4





= 51 × 110
∴ S₅₁ = 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.
Answer:
S₇ = 49, S₁₇ = 289, S₁₁ =?




2a + 6d = 14 …………… (i)





∴ 2a + 16d = 34 …………. (ii)

10d = 20
∴ d=2
Substituting the value of d in equation (i), we have
2a + 6d = 14
2a + 6 × 2 = 14
2a + 12 = 14
2a = 14 – 12
2a = 2
a = 1, d = 2, S_n =?
∴ S_n = n²

Question 10.
Show that a₁, a₂, a₃, … a₁₁.… form an AP where a₁₁ is defined as below:
i) a = 3 + 4n
ii) a = 9 – 5n
Answer:
i) a_n= 3 + 4n
a₁ = 3 + 4(1)
= 3 + 4
∴ a_n = 7
∴ a = 7
a₁ = 3 + 4n
a₂ = 3 + 4 × 2
= 3 + 8
∴ a₂ = 11
a_n = 3 + 4n
a₃ = 3 + 4 × 3
= 3 + 12
a₃ = 15
∴ a₁, a₂, a₃, ………….
7, 11, 15, ……….
d = a₂ – a₁ = 11 – 7 = 4
d = a₃ – a₂= 15 – 11 = 4
Here, the value of ‘d’ is constant.
∴ a_n = 3 + 4n forms an Arithmetic Progression.

ii) a_n = 9 – 5n
a₁= 9 – 5 × 1
= 9 – 5
∴ a₁ = 4
a₁ = 9 – 5n
a₁ = 9 – 5 × 2
= 9 – 10
∴ a₂ = -1
a_n= 9 – 5n
a₃ = 9 – 5 × 3
= 9 – 15
∴ a₃ = -6
a₁, a₂, a₃, …………… a_n
4, -1, -6, …………
d = a₂ – a₁ = -1 – 4 = -5
d=a₃ – a₂ = -6 – (-1) = -5
Here, the value of ‘d’ is constant.
∴ a_n = 9 – 5n form an Arithmetic Progression.

Question 11.
lf the sum of the first n terms of an AP is 4n – n², what is the first term (that is SI)? What is the sum of first two termš? What is the second term? Similarly. find the 3^rdrd the 10^th and the n^th terms.
Answer:
If S_n = 4n – n², then
i) S₁ = a = ?
ii) S₂ = ?
iii) a₂ = ?
iv) a₃ = ?
v) a₁₀ = ?
vi) a_n = ?
(i) S_n = 4n – n²
S₁ = 4(1) – 1²
= 4 – 1
S₁ = 3
∴ S₁ = a = 3.

(ii) S₂ = 4n – n²
S₂ = 4(2) – 2²
= 8 – 4
S₂ = 4
∴ S₂ = 4

(iii) We have S₂ = a₁ + a₂ = 4
= 3 + a₂ = 4
∴ a₂ = 4 – 3
∴a₂ = 1

(iv) S_n = 4n – n²
S₃= 4(3) – 3²
= 12 – 9
S₃ = 3
a₁ + a₂ + a₃ = 3
3 + 1 + a₃ = 3
4 + a₃= 3
∴ a₃ = 3 – 4
∴ a ₃ = -1

(v) d = a₃ – a₃ = -1 -1 = -2
a₁₀= a + 9d
= 3 + 9( – 2)
= 3 – 18
a₁₀= – 15
∴ a₁₀ = – 15

(vi) a_n = a + (n – 1)d
= 3 + (n – 1)(- 2)
= 3 + 2n + 2
∴ a_n = 5 – 2n
∴ a_n = 5 – 2n

Question 12.
Find the sum of the first 40 positive integers divisible by 6.
Answer:
6 + 12 + 18 + 24 + 40 term
Here a = 6, d = ₂ – a₁ = 12 – 6 = 6
n = 40, S₄₀ =?
\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
\(S_{40}=\frac{40}{2}[2 \times 6+(40-1) 6]\)
= 20[12 + 39 × 6]
= 20[12 + 234]
= 20 × 246
∴ S₄₀ = 4920.

Question 13.
Find the sum of first 15 multiples of 8,
Answer:
Sum of the first 15 multiples of 8?
8 + 16 + 24+ ……… 15 terms.
Here, a = 8, d = a₂ – a₁= 16 – 8 = 8
n = 15, S₁₅ =?
= 15 × 64
∴ S₁₅ = 960

Question 14.
Find the sum of the odd numbers between 0 and 50.
Answer:
Odd numbers between 0 and 50 are
1 + 3 + 5 + 7 + ……… + 49 =?
Here. a = 1, d = a₃ – a₁= 3 – 1 = 2
a_n = 49, n =?, S =?
a_n = a + (n – 1) d = 49
= 1 + (n – 1)2 = 49
1 + 2n – 2 = 49
2n – 1 = 49
2n = 49 + 1
2n = 50
= 25 × 25
∴ S₂₅ = 625

Question 15.
A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day. Rs. 250 for the second day. Rs. 300 for the third day. etc.. the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay a penalty. If he has delayed the work by 30 days?
Answer:
Penalty for the 1st Day 2nd-day 3rd day … 30^th day
Rs. 200 Rs. 250 Rs. 300 Rs.?
200 + 250 + 300 + 30 days
a = 200, d = 250 – 200 = 50, n = 20,
S₃₀ = ?
= 15[400 + 29 × 50]
= 15 [400 + 1450]
= 15 × 1850
∴ S₃₀= Rs. 27750.

Question 16.
A sum of Rs. 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize, find the value of each of the prizes.
Answer:
Let the first prize be ‘a’.
and the Second prize is a – 20 and
the third prize is 40.
a, (a – 20), (a – 40) ……….. n = 7
a = a, d = a₂ – a₁ = a – 20 – a d = -20
n = 7, S₇ =?
2a – 120 = 200
∴ 2a = 200 + 120
2a = 320
∴ Each prizes are
Rs. 160, 140, 120, 100, 80, 60, 40

Question 17.
In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the 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 Il 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?
Answer:

a = 3. d = 6 – 3 = 3, n = 12, S₁₂ = ?
= 6[6 + 11 × 3]
= 6[6 + 33]
= 6 × 39
∴ S₁₂ = 234
∴ Total number of trees from 3 sections of each class upto 12 class is 234.

Question 18.
A spiral is made up of successive semicircles, 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 shows In fig. What Is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π = \(\frac {22}{7}\))

[Hint: Length of successive semicircles is l₁, l₂, l₃, l₄ with centres at A, B, A. B respectively.]
Answer:
∴ Arithmetic Progression,
l₁, l₂, l₃, l₄, ………..

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 the fig. given below). In how many rows are the 200 logs placed and how many logs are In the top row?

Answer:
20, 19, 18, …..
a = 20, d = 19 – 20 = – 1
S_n = 200, n =?, a_n =?
∴ 400 = n(41 – n)
400 = 4n – n²
∴ n² – 41n + 400 = 0
n² – 25n – 16n + 400 = 0
n(n – 25) – 16(n – 25) = 0
(n – 25) (n – 16) =
1f n – 16 = 0 then, n = 16
∴ a_n = a + (n – 1) d
a₁₆= 20 + (16 – 1) (- 1)
= 20 + 15(- 1)
= 20 – 15
∴ a₁₆ = 5
∴ 200 logs are placed In 16 rows and there are 5 logs in the top row.

Question 20.
In a potato race, a bucket is placed at the starting point, which Is 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, given below)

A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops ¡tin the bucket, runs back to pick up the next potato, runs to the bucket to drop It in, and she continues in the same way until al) the potatoes are in the bucket. What is the total distance the competitor has to run?
(Hint: To pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 × 5 +2 × (5 + 3)]
Answer:
Total distance competitor taken to pick up the first potato = 5 + 5m
= 2 × 5m.
= 10m.
Total distance taken by compeUtor to pick up the second potato
= 5 + 3 + 3 + 5m.
= 2 × 5 + 2 × 3
= 2(5 + 3)
= 2 × 8
= 16m.
∴ 10m, 16m, 22m 10th potato
a = 10, d = 16 – 10 = 6m.
n = 10, S₁₀ = ?
= 5[20 + 9 × 6]
= 5 [20 + 54]
= 5 × 74
∴ S₁₀ = 370m.
∴ Total distance the competitor has to run to pick up 10 potatoers is 370 m.