Arithmetic Progressions Class 10

Ex 5.1, 1 teachoo.com
In which of the following situations, does the list of numbers
involved make as 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.
Taxi fare for 1 km = Rs 15
Taxi fare for 2 km=15+8
=Rs 23
Taxi fare for 3 km = 23+ 8
=Rs 31
Therefore, Series is
15, 23, 31........

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.

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

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

Ex 5.1, 2 teachoo.com
Write first four terms of the A.P. when the first term @ and the
common difference d are given as follows
{i)a=10,d=10
First term = a = 10
Common difference = d = 10
Second term = First term + Common difference
=10+10
=20
Third term = Second term + Common difference
=20+10
=30

Ex 5.1, 3 teachoo.com
For the following A.P.s, write the first term and the common
difference.
(i) 3, 1, -1, -3 ...
3,1, -1, -3...
First term =a =3
Common difference = d = Difference between 2 consecutive terms
= Second term — First Term
=1-3
= -2
Hence, a=3,d= -2

teachoo.com
Ex 5.1, 4
Which of the following are APs? If they form an A.P.
Find the common difference d and write three more terms.
(i) 2, 4, 8, 16...
2, 4, 8, 16.....

Difference of second and first term = 4 —2

=2
Difference of third and second term = 8-4

=4

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Ex 5.1, 4
Which of the following are APs? If they form an A.P. find the
common difference d and write three more terms.
(vi) 0.2, 0.22, 0.222, 0.2222 ....
0,2, 0.22, 0.222, 0.2222
Difference between second and first term = 0,22 -0.2

= 0.22 -—0.20

= 0.02
Difference between third and second term = 0.222 -0,22

= 0,222 -—0,220

= 0.002

teachoo.com

Ex 5.1, 4
Which of the following are APs? If they form an A.P. find the common
difference d and write three more terms.
(xi) a, a’, a°, a? ...
a,a’?,a°,a* ...
Difference between second and first term =a*-a

=a(a-1)
Difference between third and second term = a? — a?

=a? (a-1)

Ex5.2, 1 teachoo.com
Fill in the blanks in the following table, given that a is the first
term, d the common difference and a, the n“ term of the A.P.
a pad in| a,
7 3 8 ..
Here,a=7,d=3,n=8
Now,
a,=a+(n-1)d

Putting values

=7+(8-1) x3

=7+7X3

=7+21

=28

Ex 5.2, 2 teachoo.com
(i) Choose the correct choice in the following and justify
30% term of the A.P: 10, 7, 4, ..., is
A. 97 B. 77 C.-77 D. - 87
Given AP

10, 7, 4, ....
We have to find 30" term

So, n= 30
a=10
&d=7-10= -3

Choose the correct choice in the following and justify
11th term of the A.P. – 3, – 1/2, 2….. is
A. 28 B. 22 C. − 38 D. –48 1/2

teachoo.com
Ex 5.2, 3
In the following APs find the missing term in the boxes
(i) 2,1] ,26
Let 2"4 term =x
So, AP is
2, x, 26
Since it is an AP,
Common difference is same
«. Second term — First term = Third term — Second term
x-2=26-x
X+X=2+26

In the following APs find the missing term in the boxes
(ii) ⎕, 13, ⎕, 3

In the following APs find the missing term in the boxes
(iii) 5, ⎕, ⎕, 9 1/2

In the following APs find the missing term in the boxes
(iv) – 4, ⎕, ⎕, ⎕, ⎕, 6

In the following APs find the missing term in the boxes
(v) ⎕, 38, ⎕ , ⎕, ⎕, − 22

Ex 5.2, 4 teackoo.com
Which term of the A.P. 3, 8, 13, 18, ... is 78?
Given AP

3, 8, 13,18, ...
We need to find which term is 78
Hence,

a, = 78

a=3

d=8-3=5
Putting these in formula

a, =at(n-1)d

Ex5.2,5 teachoo.com
Find the number of terms in the following AP
(i) 7, 13, 19, ..., 205
Given AP

7, 13, 19, ... 205
Here,

a=7

d=13-7=6

a, = 205
We need to find n

a,=at+(n-1)d

205 =7+(n-1)x6

Find the number of terms in the following AP
(ii) 18, 151/2, 13,……….,-47

Ex 5.2, 6 teachoo.com
Check whether - 150 is a term of the A.P. 11, 8, 5, 2, ...
Given AP
11, 8, 5, 2, ....
We need to check whether —150 is a term of AP
Lets assume it is n term of AP
So, a,=—150
Also, a= 11,
d=8-11=-3
Now,
a, =at(n-1)d

Ex 5.2, 7 teachoo.com

Find the 31° term of an A.P. whose 11" term is 38 and the

16" term is 73

We know that
a,=at(n-1)d

Given 11™ term is 38 Given 16" term is 73
a,,=at+(11-1)d ayg=at(16-1)d
38=a+(11-1)}d ayg=at15d
38=a+10d 73=a+15d
38-10d =a 73-15d=a
a=38-10d ...(1) a=73-15d__...(2)

Ex 5.2, 8 teachoo.com
An A.P. consists of 50 terms of which 3 term is 12 and the last
term is 106. Find the 29" term
We know that
a,=a+(n—1)d
Given 3 term is 12 Given last term is 106
a,=a+(3-1)d Last term = 50" term = ac, = 106
12=a+2d Now,
12-2d =a aso =a +(50-1)d
azw-24 ~-(2) 106 =a+49d
106 - 49d=a
a= 106 - 49d (2)

Ex 5.2, 9 teachoo.com
If the 34 and the 9" terms of an A.P. are 4 and — 8 respectively.
Which term of this A.P. is zero.
We know that
a,=a+(n-1)d
Given 3 term is 4 Given 9" term is -8
a,=a+(3-1)d a,=a+(9—1)d
4=at+2d -8=a+8d
4-2d=a -8-8d=a
a=4-2d (1) a= —8d-8 --(2)

teachoo.com

Ex 5.2, 10
If 17" term of an A.P. exceeds its 10" term by 7. Find the
common difference.
We know that

a,=a+(n-1)d
So, a,,=a+(17-1)d

ay, =a+16d (1)
Also,

ay =at+(10—-1)d

ay = a+ 9d (2)

Ex 5.2, 11 teachoo.com
Which term of the A.P. 3, 15, 27, 39, ... will be 132 more than its
54" term?
Given AP

3, 15, 27, 39...
First we need to calculate 54" term
Here,a=3

d=15-3=12

n=54

Ex 5.2, 12 teachoo.com
Two APs have the same common difference. The difference between
their 100" term is 100, what is the difference between their
1000" terms?
First AP Second AP
We know that We know that
a, =a+(n-1)d b,=b+(n-1}d
So, 100" term will be So, 100" term will be
Aino = at (100-1) d biog = b + (100 —1)d
aio) = a+ 99d bigg = b + 99d
So, 100" term of 1° AP=a+99d | So, 100" term of 2" AP = b + 99d

Ex 5.2, 13 teachoo.com
How many three digit numbers are divisible by 7
Numbers divisible by 7 are
7, 14, 21, 28, o....--
Lowest 3 digit number Highest 3 digit number,
100 2142 oe = 1422
7 7 7 7
at =a oe = 1424
7 7 7 7
10? = 14t 27 = 1428
7 7 7 7
105 45 4 5142
7 7
So lowest number is 105 Highest number is 994

Ex 5.2, 14 teachoo.com
How many multiples of 4 lie between 10 and 250?
Multiple of 4 are

4,8, 12, 16, ....
For Lowest multiple For highest multiples

10 _ 42 250 _ gp2

4 4 4 4

u_ 23 249 _ 691

4 4 4 4

¥u3 248 = 62

4 4
-. Lowest multiple = 12 * Highest multiple = 248

Ex 5.2, 15 teachoo.com
For what value of n, are the n terms of two APs 63, 65, 67,... and
3, 10, 17, ... equal

Let’s find n® term of both APs

Ex 5.2, 16 teachoo.com
Determine the A.P. whose third term is 16 and the 7" term exceeds
the 5" term by 12
We know that
a,=a+(n-1)d
Let’s find the 3, 5 and 7" term
a; as ay
a,=a+(3-1)d a;za+(5—1)d ay=a+(7-1)d
16= a+2d =atdd =a+é6d
a+2d=16 ..(1)

Ex 5.2, 17 teachoo.com
Find the 20" term from the last term of the A.P. 3, 8, 13, ..., 253
Given AP

3, 8, 13, ..., 253
Now,

20" term from last term of AP 3, 8, 13, ...,

= 20" term of AP 253, 248, 243, ...., 8, 3

Thus, our AP is

253, 248, 243, ...., 8,3

teachoo.com

Ex 5.2, 18
The sum of 4" and 8" terms of an A.P. is 24 and the sum of the
6" and 10" terms is 44. Find the first three terms of the A.P.
We know that

a,=a+(n-1)d
So, a,=a+(4-1)d

a,=a+3d (1)
Also, ag =a + (8—1)d

ag=a+7d ...(2)

teachoo.com

Ex 5.2, 19
Subba Rao started work in 1995 at an annual salary of Rs 5000 and
received an increment of Rs 200 each year. In which year did his
income reach Rs 7000?
Salary in 1995 (First year) = Rs 5000
Salary in 1996 (second year) = 5000 + 200 = Rs 5200
Salary in 1997 (Third year) = 5200 + 200 = Rs 5400
So, the series is

5000, 5200, 5400, ....
Since difference is same, it is an AP
Common difference = d= 200

Ex 5.2, 20 teachoo.com
Ramkali saved Rs 5 in the first week of a year and then increased her
weekly saving by Rs 1.75. If in the n‘* week, her weekly savings
become Rs 20.75, find n.
Saving made first week = Rs 5
Saving made in second week = Rs 5 + 1.75 = Rs 6.75
Saving made in third week = 6.75 +1.75 =Rs 8.50
So, the series is

5, 6.75, 8.50 ......
Since difference is same, it is an AP

First Term =a=5

Common difference = d= 1.75

teachoo.com

Ex 5.3,1
Find the sum of the following APs.
(i) 2, 7, 12,...., to 10 terms.

2,7, 12,....,to 10 terms
We know that

Sum of AP=~ (2a +(n—1)d)
Heren=10, a=2,

&d=7-2=2

Putting these in formula,
Sum =F (2a+(n-1) d)

Find the sum of the following APs.
(ii) −37, −33, −29 ,…, to 12 terms

Find the sum of the following APs.
(iii) 0.6, 1.7, 2.8 ,…….., to 100 terms

Find the sum of the following APs.
(iv) 1/15, 1/12, 1/10 ,………, to 11 terms

teachoo.com
Ex 5.3, 2
Find the sums given below
(7+ 10; +14 t ce + B84
7+1054+14+4--84
Here, a=7
1
d=10 77 7
21

=>7 7

_21-14_7

~ 20 72
Also, last term = [= 84

Find the sums given below
(ii) 34 + 32 + 30 + ……….. + 10

Find the sums given below
(iii) − 5 + (−8) + (−11) + ………… + (−230)

teachoo.com

Ex 5.3, 3
In an AP
(i) Given a = 5, d= 3, a, = 50, find n and S,.
Given a=5,d=3,a,=50
We know that

a,=at+(n-1)d
Putting values

50=5+(n-1)x3

50=5+3n-3

S0=2+3n

S0-2=3n

48 = 3n

In an AP
(ii) Given a = 7, a13 = 35, find d and S13.

In an AP
(iii) Given a12 = 37, d = 3, find a and S12.

Given a3 = 15, S10 = 125, find d and a10.

In an AP
(v) Given d = 5, S9 = 75, find a and a9.

In an AP
(vi) Given a = 2, d = 8, Sn = 90, find n and an.

In an AP
(vii) Given a = 8, an = 62, Sn = 210, find n and d.

In an AP
(viii) Given an = 4, d = 2, Sn = −14, find n and a.

In an AP
(ix) Given a = 3, n = 8, S = 192, find d.

In an AP
(x) Given 𝑙 = 28, S = 144 and there are total 9 terms. Find a.

Ex 5.3, 4 teachoo.com
How many terms of the AP. 9, 17, 25 ... must be taken to give asum
of 636?
Given AP is 9, 17, 25, .....
Here, a=9

d=17-9=8

& Sum = S, = 636

We need to find n
We know that

Sum = 5 (2a +(n-1) d)

Ex 5.3,5 teachoo.com
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.
Given
a=5,
7=45
& S,=Sum = 400
Since last term is given, we can use formula
n
S,=2(a+
Putting values in formula
400 = = (5 + 45)

Ex 5.3, 6 teachoo.com
The first and the last term of an AP are 17 and 350 respectively.
If the common difference is 9, how many terms are there and what is
their sum?
Given a=17,/=350,d=9
We need to find how many terms i.e. n
and their sum i.e. S,
We know that
a, =a+(n-1)d
Putting a=17,a, = 1=350,d=9
350 =17+(n-1)x9

Ex 5.3, 7 teachoo.com
Find the sum of first 22 terms of an AP in which d = 7 and
224 term is 149.
Given

22"4 term = a), = 149

Common difference = d =7
We know that

a,=at(n-1)d
Putting n=22,d=7,a,=149

149 =a+(22-1) x7

149=a+21x7

149 =a+147

Ex 5.3, 8 teachoo.com
Find the sum of first 51 terms of an AP whose second and third
terms are 14 and 18 respectively.
We know that
a, =a+{n—1)d
Given 2" term is 14 Given 3 term is 18
a,=a+(2-1)d a,=a+(3-1)d
14=a+d 18=a+2d
14-d=a 18-2d=a
a=14-d_...(1} a= 18-2d ...(2)
From (1) & (2)
14-d= 18-2d

Ex 5.3,9 teachoo.com
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.
We know that
$,=5 (2a + (n— 1a)
Sum of first 7 terms = 49 Sum of first 17 terms = 289
S,=2(2a + (n—1)d) S7= (@a + (17 - 1d)
49 =2 (2a + (7 —1)d) 289 = 2 (2a + (17-1) d)
7 17
49 = > (2a + 6d) 289 =— (2a + 16d)
OX? = 2a+6d 280X? _ 2a +16d
7 17
14 = 2a + 6d 34=2a+16d
14 -6d 34 - 16d
——=a ——=a
2 2
a=7-3d (1) a=17-8d ...(2}

Ex 5.3, 10 teachoo.com
Show that a,, a,...,@,, ... form an AP where a, is defined as below
(i) a, = 3 + 4n. Also find the sum of first 15 terms
Since a, = 3 + 4n
Taking n=1 Taking n = 2 Taking n = 3
a,=3+4x1 a=3+4x2 a,=3+4x3
a=3+4 a,=3+8 a,=3+12
a=7 a,=11 a,=15
Hence, series is 7, 11, 15, .....
Since difference is same, it is an AP
Common difference = d=4

Show that a1, a2 … , an , … form an AP where an is defined as below
(ii) an = 9 − 5n . Also find the sum of first 15 terms in each case

Ex5.3, 11 teachoo.com
lf the sum of the first n terms of an AP is 4n - n2, what is the first
term (that is $,)? What is the sum of first two terms? What is the
second term? Similarly find the 3%, the 10" and the n“ terms.
Given
S,=4n—n?
Taking n=1 Taking n = 2 inS,
$,=4x 1-17 S,=4x 2 — 2?
=4-1 S,=8-4
=3 S,=4
- Sum of first term of APis3 | «Sum of first 2 terms is 4

Ex 5.3, 12 teachoo.com
Find the sum of first 40 positive integers divisible by 6.
Positive integers divisible by 6 are

6, 12, 18, 24.....
Since difference is same, it is an AP
We need to find sum of first 40 integers
We can use formula

Sy=5 (2a + (n-1) d)
Here,n=40,

a=6

&d=12-6 =6

Ex 5.3, 13 teachoo.com
Find the sum of first 15 multiples of 8.
Multiples of 8 are

8, 16, 24.,...
Since difference is same, it is an AP
We need to find sum of first 15 multiples
We use formula

Sn= 5 (2a +(n-1)d)
Here, n= 15,

a=8

&d=16-8=8

Ex5.3, 14 teachoo.com
Find the sum of the odd numbers between 0 and 50.
Odd numbers between 0 to 50 are
1, 3,5, 7, ...., 49

Here,

First term =a=1

Last term =/= 49
There are 25 such terms
So,n=25
We need to find sum

Ex5.3, 15 teachoo.com
A contract on 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 as
penalty, if he has delayed the work by 30 days.
Penalty for 1st day = 200
Penalty for 2"? day = 250
Penalty for 3 day = 300
Hence, the series is

200, 250, 300, ....

Ex 5.3, 16 teachoo.com
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.
<5-45
_ > *
Let the 1° prize be a r .\
So, 2" prize = a— 20 ‘ u
The 3" prize = (a—20)—20=a-40 -
And so on
So, the series will be
a,a-20,a-—40, ....
Since difference is same, it is an AP

Ex 5.3, 17 teachoo.com
In aschool, 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 | 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? 260
Trees planted by class 1% =1 x 3=3

Trees planted by class 24 =2 x 3=6

Trees planted by class 3° =3 x 3=9

Trees planted by class 12" = 12 x 3 = 36

Ex 5.3, 18 teachoo.com
A spiral is made up of successive semicircles, with centres
alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm,
1.5m, 2.0 cm,......... as shown in figure. What is the total length of
sucha spiral made up of thirteen consecutive semicircles?
22
{take m = —) r=15cem \
\
\
a
1
Here we have to find total length, i.e., ‘
total circumference of all sernicircles ee /
We know that
Circumference of circle = 2nr
So, Circumference of semicircle = ; x 2m =r

teachoo.co
Ex 5.3, 19 ma
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. In how
many rows are the 200 logs placed and how many logs are in the top
row?
(err errreeeeerr rrr y/
©0O0SOOC0CCOCOOCCCOe
Number of logs in 1% row = 20
Number of logs in 2" row = 19
Number of logs in 3% row = 18
Hence the series is
20, 19, 18, ........
Since difference is same, it is an AP

Ex 5.3, 20 teachoo.com
In a potato race, a bucket is placed at the starting point, which is 5 m
from the first potato and other potatoes are placed 3 m apart ina
straight line. There are ten potatoes in the line.
ogo ®t
A 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 she continuesin
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 the second potato, the total
distance (in metres} run by a competitor is 2 x 5 +2 x (5+ 3}]
Distance run to pick first potato = 5 m each side

=5mx2

= 10 m both sides

teachoo.com
Example 1
3 1 -1 -3 . .
For the AP: og? oge write the first term a & the common
difference d
3 1-1 -3
27.2’? 27 27°"
: 3
First term =a = 3
Common difference = d = Difference between 2™ and 1% term
_1_ 3
“2 2
_1-3
"2

Example 2 teachoo.com
Which of the following list of numbers does form an AP? If they
form an AP, write the next two terms :
(i) 4,10, 16, 22 eee
Difference between 2™ and 1% term

=10-4

=6
Difference between third and second term

= 16-10

=6

Example 3 teachoo.com
Find the 10th term of the AP: 2, 7, 12,...
Given AP
2,7, 12, crscserseee

We have to find 10" term
So,n= 10
Also,a=2&d=7-2=5
Putting these value in formula

a, =a+(n—1)d

ay) = 2 + (10-1) (5)

ayy =24+9x5

Example 4 teachoo.com
Which term of the AP : 21, 18, 15, .. . is -81? Also, is any term 0?
Give reason for your answer.
Given AP
21, 18, 15, cooceeessesee
We need to find which term is — 81.
Thus,
a,= —-81
&a=21&d =18-21=-3
We find to find n

Example 5 teachoo.com
Determine the AP whose 3% term is 5 and the 7" term is 9.
We know that
a, =a+(n—-1)d
Given 3" term is 5 Given 7" term is 9
a,=a+(3-1)d a,=at+(7-1)d
5=at+2d 9=a+6d
a=5-2d --(1} a=9-6d_ ~(2)
From (1) & (2)
§-2d=9-6d
6d-2d=9-5

teachoo.com

Example 6
Check whether 301 is a term of the list of numbers 5, 11, 17, 23,...
Given A.P.

5,11, 17, 23,
We need to cheek whether 301 is a term of AP
Lets assume it is n term of AP
So, a, = 301
Also,a=5 &d=11-5=6
Now,

a,=at(n-1)d

Example 7 teachoo.com
How many two-digit numbers are divisible by 3?
Numbers divisible by 3 are
3,6, 9,12, reece
Lowest two digit number divisible by 3 is 12
Highest two digit number divisible by 3
We know that = =33
«. Highest two digit number divisible by 3 is 99
So, the series starts with 12 and ends with 99.
Difference between numbers is 3

Example 8 teachoo.com
Find the 11 term from the last term (towards the first term) of the
AP : 10, 7, 4, ..., - 62.
Given AP

10, 7, 4, vss, 62
Now,

11 term from last term of AP 10, 7, 4, ....

= 11" term of AP -62, -59, —56, ....7, 10

Thus, our AP is

-62, -59, -56, ....7, 10

Example 9 teachoo.com
Asum of Rs 1000 is invested at 8% simple interest per year. Calculate
the interest at the end of each year. Do these interests form an AP? If
so, find the interest at the end of 30 years making use of this fact.
We know that

Interest = PRRXT

100

Here P = Principal, R = Rate , T = Time
Given,

P= 1000, R =8% &T=T years

1000x8x1

Interest at the end of 1% year = 00 = Rs 80

Example 10 teachoo.com
In a flower bed, there are 23 rose plants in the first row, 21 in the
second, 19 in the third, and so on. There are 5 rose plants in the last
row. How many rows are there in the flower bed?
Given, Rose plants in 1%, 2°¢ ,, 3" rows are:

23,21, 19, caoreeuey 5 SQ CF
Since Difference between consecutive terms is same,
it is an AP
We have to find number of row in the flower bed
Lets assume there are n rows in the flower bed

Example 11 teachoo.com
Find the sum of the first 22 terms of the AP : 8, 3,-2,...
Given
8, 3, 2, sree
We have to find sum of first 22 terms .
Here, n= 22,
a=8
d=3-8=-5
Putting there values in formula,
Sum =$[2a +(n-1)d]

Example 12 teachoo.com
If the sum of the first 14 terms of an AP is 1050 and its first term is
10, find the 20" term.
Here, Sum of the first 14 terms of an AP is 1050.
So, Si, = 1050
Also, a=10 &n=14
We know that
Sum = > [Za + (n — 1)d]
Putting values
1050 =— [2 x (10) + (14 - 1d]
1050 = 7[20 + 13d]

Example 13 teachoo.com
How many terms of the AP : 24, 21, 18, ... must be taken so that
their sum is 78?
Given AP ;
24, 21, 18,.........
Here,
a=24
d=21-24= -3
Also, given Sum = 78
S, =78
We have to find value of n

Example 14 (Method 1) teachoo.com
Find the sum of
(i) the first 1000 positive integers
Positive integers start from 1
First 1000 positive integers are
1, 2, 3, 4, 1... 1000
This is an AP with
First term =a=1
Common Difference=d=1
Number of terms = n = (1000 - 1) + 1 = 1000

Example 14
Find the sum of :
(ii) the first n positive integers

teachoo.com
Example 15
Find the sum of first 24 terms of the list of numbers whose n‘*
term is given by a, = 3 + 2n
Given
a,=3+2n

Now,
1% term = a, = 3 + 2 (1)

a=3+2

a=5
2™4 term = a, = 3 + 2(2)

=3+4=7

Example 16 teachoo.com
A manufacturer of TV sets produced 600 sets in the third year and 700
sets in the seventh year. Assuming that the production increases
uniformly by a fixed number every year , find:
{i) the production in the 1st year
Since production increases by a fixed number every year, it is an AP
Given 3" year production is 600 | Given 7“ year production is 700
So, a, = 600 So, a, = 700
We know We know

a,=a+(n—1)d a,=a+(n-1)d

a,=a+(3-1)d aj=a+(7-1}d

600 =a+2d 700 = a+ 6d

600 -2d=a 700 —6d =a

a= 600 - 2d (1) a= 700 - 6d ...(2}

Ex 5.4, 1 (Optional) feachoo.com
Which term of the AP : 121, 117, 113, . . ., is its first negative term?
(Hint : Find n for a, < 0]
Given AP is

121, 117, 113.,......
Here, a= 121

d=117-121=-4
Let the n term of the AP be its first negative term.

a, <0

at+(n-1)d<0

Putting a= 121 and d= -4

Ex 5.4, 2 (Optional) teachoo.com
The sum of the third and the seventh terms of an AP is 6 and their
product is 8. Find the sum of first sixteen terms of the AP.
We know that
n“ term of an AP is
a, =a+(n-1)d

Hence,

3" term of AP = a, =a+2d
and 7" term of AP = a, =a+ 6d
Given
Sum of third & seventh terms is 6

Ex 5.4, 3 (Optional) - Introduction teachoo.com
A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease
uniformly in length from 45 cm at the bottom to 25 cm at the top. If
the top and the bottom rungs are 25 m apart, what is the length of
the wood required for the rungs?
Introduction 5 en]
Let there be a Ladder
which has 5 rungs 5 cm apart 20cm
The top and the bottom rungs are 20 cm apart.
We say,

Number of rungs = “ =4

Ex 5.4, 4 (Optional) - Introduction teachoo.com
The houses of a row are numbered consecutively from 1 to 49.
Show that there is a value of x such that the sum of the numbers of
the houses preceding the house numbered x is equal to the sum of
the numbers of the houses following it. Find this value of x.
[Hint :$,_ 4 =S49—- $,]
Introduction:
Let 5 houses numbered from 1 to 5 be arranged in a row

1 2 3 4 =5
Now, let x = 3
Then

Ex5.4, 5 (Optional) teachoo.com
A small terrace at a football ground comprises of 15 steps each of
which is 50 m long and built of solid concrete. Each step has a rise of
: m and a tread of 5 m. (see Fig. 5.8). Calculate the total volume of
concrete required to build the terrace.
[Hint: Volume of concrete required to build the first step = : x ; x
50 m?]
Given ,
im ls
Number of stepsn = 15, 2 ‘€
S)
—m oS
Length of Steps =50 m4 id
Rise in each step =2 m
Tread of each other step =5 m

India is competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in 6th year and 22600 in 9th year.
Based on the above information, answer the following questions:
Question 1
Find the production during first year.
Question 2
Find the production during 8
th
year.
Question 3
Find the production during first 3 years.
Question 4
In which year, the production is Rs 29,200.
Question 5
Find the difference of the production during 7th year and 4th year.

Your friend Veer wants to participate in a 200m race. He can currently run that distance in 51 seconds and with each day of practice it takes him 2 seconds less. He wants to do in 31 seconds
Question 1
Which of the following terms are in AP for the given situation
(a) 51, 53, 55….
(b) 51, 49, 47….
(c) –51, –53, –55….
(d) 51, 55, 59…
Question 2
What is the minimum number of days he needs to practice till his goal is achieved
(a) 10
(b) 12
(c) 11
(d) 9
Question 3
Which of the following term is not in the AP of the above given situation
(a) 41
(b) 30
(c) 37
(d) 39
Question 4
If n
th
term of an AP is given by
a
n
= 2n + 3 then common difference of an AP is
(a) 2
(b) 3
(c) 5
(d) 1
Question 5
The value of x, for which 2x, x+ 10, 3x + 2 are three consecutive terms of an AP
(a) 6
(b) –6
(c) 18
(d) –18

Your elder brother wants to buy a car and plans to take loan from a bank for his car. He repays his total loan of Rs 1,18,000 by paying every month starting with the first instalment of Rs 1000. If he increases the instalment by Rs 100 every month , answer the following:
Question 1
The amount paid by him in 30
th
installment is
(a) 3900
(b) 3500
(c) 3700
(d) 3600
Question 2
The amount paid by him in the 30 installments is
(a) 37000
(b) 73500
(c) 75300
(d) 75000
Question 3
What amount does he still have to pay offer 30th installment?
(a) 45500
(b) 49000
(c) 44500
(d) 54000
Question 4
If total installments are 40 then amount paid in the last installment?
(a) 4900
(b) 3900
(c) 5900
(d) 9400
Question 5
The ratio of the 1
st
installment to the last installment is
(a) 1 : 49
(b) 10 : 49
(c) 10 : 39
(d) 39 : 10

Chapter 5 Class 10 - Arithmetic Progressions
- MCQ Worksheet 1
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Chapter 5 Class 10 - Arithmetic Progressions
- Mix Questions Wsorksheet 1
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Chapter 5 Class 10 - Arithmetic Progressions
- Assertion and Reasoning
Worksheet 1
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Chapter 5 Class 10 - Arithmetic Progressions
- Case Based Question
Worksheet 1
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