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SEBA Class 10 Mathematics Chapter 5 Arithmetic Progressions
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Arithmetic Progressions
Chapter – 5
| Exercise 5.1 |
1. In which of the following situations, does the list of numbers in- volved 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.
Ans: It can be observed that
Taxi fare for 1st 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
Clearly 15, 23, 31, 39 … forms an A.P. because every term is 8 more
than the preceding term.
(ii)The amount of air present in a cylinder when a vacuum pump removes ¼ 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 Rs 50 for each subsequent metre.
Ans: 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 Clearly, 150,
200, 250, 300 … forms an A.P. because every term is 50 more
than the preceding term.
(iv) The amount of money in the account every year, when Rs. 10000 is deposited at compound interest at 8% per annum.
2. Write first four terms of the A.P. when the first term a and the common difference d are given as follows.
(i) a = 10, d = 10
Ans: a = 10, d = 10
Let the series be a1, a2, a3, a4, a5
a1 = a = 10
a2 = a1 + d = 10 + 10 = 20
a3 = a2 + d = 20 + 10 = 30
a4 = a3 + d = 30 + 10 = 40
a5 = a4 + d = 40 + 10 = 50
Therefore, the series will be 10, 20, 30, 40, 50 … First four terms of this A.P. will be
10, 20, 30, and 40.
(ii) a = −2, d = 0
Ans: Let the series be a1, a2, a3, a4 …
a1 = a = −2
a2 = a1 + d = − 2 + 0 = −2
a3 = a2 + d = − 2 + 0 = −2
a4 = a3 + d = − 2 + 0 = −2
Therefore, the series will be −2, −2, −2, −2 …
First four terms of this A.P. will be −2, −2, −2 and −2.
(iii) a = 4, d = −3
Ans: Let the series be a1, a2, a3, a4 …
a1 = a = 4
a2 = a1 + d = 4 − 3 = 1
a3 = a2 + d = 1 − 3 = −2
a4 = a3 + d = − 2 − 3 = −5
Therefore, the series will be 4, 1, −2 −5 …
First four terms of this A.P. will be 4, 1, −2 and −5.
(iv) a = −1, d = ½ Let the series be a1, a2, a3, a4..
Ans:
(v) a = −1.25, d = −0.25
Ans: Let the series be a1, a2, a3, a4 …
a1 = a = −1.25
a2 = a1 + d = − 1.25 − 0.25 = −1.50
a3 = a2 + d = − 1.50 − 0.25 = −1.75
a4 = a3 + d = − 1.75 − 0.25 = −2.00
Clearly, the series will be 1.25, −1.50, −1.75, −2.00 …
First four terms of this A.P. will be −1.25, −1.50, −1.75 and −2.00.
3. For the following A.P.s, write the first term and the common difference.
(i) 3, 1, − 1, − 3 …
Ans: 3, 1, −1, −3
Here, first term, a = 3
Common difference, d = Second term − First term
= 1 − 3 = −2
(ii) −5, −1, 3, 7.
Ans: a = – 5 , d = 4
(iv) 0.6, 1.7, 2.8, 3.9
Ans: a = 0.6 , d = 1.1
4. Which of the following APs? If they form an AP, find the common difference d and write three more terms.
(i) 2, 4, 8, 16…
Ans: Not an AP because t2 – t1 = 2 and t3 – t – 2 = 8 – 4 = 4
i.e .. , t2 – t1 ≠ t3 – t2
(iii) – 1.2, -3.2, -5.2, -7.2, …
Ans: It is an AP.
a = -1.2, d = -2
Ts-7.2-2 = -92,t6 = – 11.2, t, = – 13.2
(iv) -10, -6, -2, 2,…
Ans: It is an AP.
a = -10, d = 4
t5 = 6, t6 = 10, t7, = 14
(vi) 0.2, 0.22, 0.222, 0.2222, ..
Ans: It is not an AP.
Here, t2, -= t2 0.22 – 0.222 – 0.02 = 0.002…
(vii) 0, -4, -8, – 12
Ans: It is an AP.
a = 0, d = -4
t5 = – t6 = -20, t7, = -24
(ix) 1, 3, 9,27.
Ans: Not an AP
Here, t2 – t1 ≠ t3 – t2
(x) a, а2, а3, α4
Ans: It is an AP. First term = a , common differences = 0
T5 = 5 , a t6 = 6, a = 6a, t7 = 7a
(xi) a, a2, a3 a4
Ans: a, a2 a3, a4……
Ans: Not an AP if a ≠ 1. Here t2 – t1 = a = a(1 – a )
T3 – t2 = a3 – a2(1 – a)
T3 – t2 ≠ t2 – when a ≠ 1
It will be an AP if a = 1
Hence, the given sequence is an AP only when a = 1 In this
case, First term = 1, common difference = 1
(xiv) 12,32,72
(xv) 12,52,72,73…
| Exercise 5.2 |
1. Fill in the blanks in the following table, given that a is the first term, d the common difference and a the nth of the A.P.
| a | d | n | an | |
| (i) | 7 | 3 | 8 | ….. |
| (ii) | − 18 | ….. | 10 | 0 |
| (iii) | ….. | − 3 | 18 | − 5 |
| (iv) | − 18.9 | 2.5 | ….. | 3.6 |
| (v) | 3.5 | 0 | 105 | ….. |
Ans: a = 7, d = 3, n = 8, an = ?
We know that,
For an A.P. an = a + (n − 1) d
= 7 + (8 − 1) 3
= 7 + (7) 3
= 7 + 21 = 28
Hence, an = 28
(ii) Given that a = −18, n = 10, an = 0, d = ?
We know that,
an = a + (n − 1) d
0 = − 18 + (10 − 1) d
18 = 9d
D = 18/9 = 2
Hence, common difference, d = 2
(iii) Given that d = −3, n = 18, an = −5
We know that, an = a + (n − 1) d
−5 = a + (18 − 1) (−3)
−5 = a + (17) (−3)
−5 = a − 51
a = 51 − 5 = 46
Hence, a = 46
(iv) a = −18.9, d = 2.5, an = 3.6, n = ?
We know that, an = a + (n − 1) d
3.6 = − 18.9 + (n − 1) 2.5
3.6 + 18.9 = (n − 1) 2.5
22.5 = (n − 1) 2.5
Hence, n = 10
(v) a = 3.5, d = 0, n = 105, an = ?
Ans: We know that, an = a + (n − 1) d
an = 3.5 + (105 − 1) 0
an = 3.5 + 104 × 0
an = 3.5
Hence, an = 3.5
2. Choose the correct choice in the following and justify
(i) 30th term of the A.P: 10, 7, 4, …, is
Ans: Given that
A.P. 10, 7, 4, …
First term, a = 10
Common difference, d = a2 − a1 = 7 − 10 = −3
We know that, an = a + (n − 1) d
a30 = 10 + (30 − 1) (−3)
a30 = 10 + (29) (−3)
a30 = 10 − 87 = −77
Hence, the correct answer is C.
(ii) Given that, A.P.
First term a = −3
Common difference, d = a2 − a1
3. In the following APs find the missing term in the boxes.
Ans: (i) For this A.P., a = 2 and a3 = 26
We know that, an = a + (n − 1) d
a3 = 2 + (3 − 1) d
26 = 2 + 2d
24 = 2d d = 12
a2 = 2 + (2 − 1) 12
= 14
(ii) Therefore, 14 is the missing term.
For this A.P., a2 = 13 and a4 = 3
We know that, an = a + (n − 1) d
a2 = a + (2 − 1) d
13 = a + d …………………………(I)
a4 = a + (4 − 1) d
3 = a + 3d ………………………..(II)
On subtracting (I) from (II), we obtain −10 = 2d
d = −5
(iv) Ans:
(v) Ans: For this A.P., a = −4 and a6 = 6
We know that, an = a + (n − 1) d
a6 = a + (6 − 1) d
6 = − 4 + 5d
10 = 5d
d = 2
a2 = a + d = − 4 + 2 = −2
a3 = a + 2d = − 4 + 2 (2) = 0
a4 = a + 3d = − 4 + 3 (2) = 2
a5 = a + 4d = − 4 + 4 (2) = 4
Therefore, the missing terms are −2, 0, 2, and 4 respectively.
4. Which term of the A.P. 3, 8, 13, 18, … is 78?
Ans: 3, 8, 13, 18,For this A.P., a = 3 and d = a2 − a1 = 8 − 3 = 5
Let nth term of this A.P. be 78.
an = a + (n − 1) d
78 = 3 + (n − 1) 5
75 = (n − 1) 5
(n − 1) = 15
n = 16
Hence, 16th term of this A.P. is 78.
5. Find the number of terms in each of the following A.P.
(i). 7, 13, 19, …, 205
Ans: 7, 13, 19, …, 205
For this A.P., a = 7 and d = a2 − a1 = 13 − 7 = 6
Let there are n terms in this A.P. an = 205
We know that an = a + (n − 1) d
Therefore, 205 = 7 + (n − 1) 6
198 = (n − 1) 6
33 = (n − 1)
n = 34
Therefore, this given series has 34 terms in it.
6. Check whether − 150 is a term of the A.P. 11, 8, 5, 2, …
For this A.P., a = 11 and d = a2 − a1 = 8 − 11 = −3 Let −150 be the nth term of this A.P. We know that,
7. Find the 31st term of an A.P. whose 11th term is 38 and the 16th term is 73.
Ans: Given that, a11 = 38 and a16 = 73
We know that, an = a + (n − 1) d
a11 = a + (11 − 1) d
38 = a + 10d ………………………..(1)
Similarly, a16 = a + (16 − 1) d
73 = a + 15d ………………………….(2)
On subtracting (1) from (2), we obtain
35 = 5d
d = 7
From equation (1),
38 = a + 10 × (7)
38 − 70 = a
a = −32
a31 = a + (31 − 1) d
= − 32 + 30 (7)
= − 32 + 210
= 178
Hence, 31st term is 178.
8. An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term
Ans: Given that, a3 = 12 and a50 = 106
We know that, an = a + (n − 1) d
a3 = a + (3 − 1) d
12 = a + 2d …………………………..(I)
Similarly, a50 = a + (50 − 1) d
106 = a + 49d ………………………(II)
On subtracting (I) from (II), we obtain
94 = 47d
d = 2
From equation (I), we obtain
12 = a + 2 (2)
a = 12 − 4 = 8
a29 = a + (29 − 1) d
a29 = 8 + (28)2
a29 = 8 + 56 = 64
Therefore, 29th term is 64.
9. If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively. Which term of this A.P. is zero?
Ans: Given that, a3 = 4 and a9 = −8
We know that, an = a + (n − 1) d
a3 = a + (3 − 1) d
4 = a + 2d ……………………(I)
a9 = a + (9 − 1) d
−8 = a + 8d …………………(II)
On subtracting equation (I) from (II), we obtain
−12 = 6d
d = −2
From equation (I), we obtain
4 = a + 2 (−2)
4 = a − 4
a = 8
Let nth
th term of this A.P. be zero.
an = a + (n − 1) d
0 = 8 + (n − 1) (−2)
0 = 8 − 2n + 2
2n = 10
n = 5
Hence, 5th term of this A.P. is 0
10. The 17th term of an A.P. exceeds its 10th term by 7. Find the common difference.
Ans: We know that,
For an A.P., an = a + (n − 1) d
a17 = a + (17 − 1) d
a17 = a + 16d
Similarly, a10 = a + 9d
It is given that a17 − a10 = 7
(a + 16d) − (a + 9d) = 7
7d = 7
d = 1
Therefore, the common difference is 1.
11. Which term of the A.P. 3, 15, 27, 39, … will be 132 more than its 54th Term?
Ans: Given A.P. is 3, 15, 27, 39, …
a = 3 d = a2 − a1 = 15 − 3 = 12
a54 = a + (54 − 1) d
= 3 + (53) (12)
= 3 + 636 = 639
Now a54 +132 = 639 + 132 = 771
We have to find the term of this A.P. which is 771.
Let nth term be 771.
an = a + (n − 1) d
771 = 3 + (n − 1) 12
768 = (n − 1) 12
(n − 1) = 64
n = 65
Therefore, the 65th term was 132 more than the 54th term.
12. Two APs have the same common difference. The difference between their 100th term is 100, what is the difference between their 1000th terms?
Ans: Let the first term of these A.P.s be a1 and a2 respectively and the common difference of these A.P.s be d.
For first A.P.,
a100 = a1 + (100 − 1) d
= a1 + 99d
a1000 = a1 + (1000 − 1) d
a1000 = a1 + 999d
For second A.P.,
a100 = a2 + (100 − 1) d
= a2 + 99d
a1000 = a2 + (1000 − 1) d
= a2 + 999d
Given that, difference between
100th term of these A.P.s = 100 Therefore,
(a1 + 99d) − (a2 + 99d) = 100 a1 − a2 = 100 ……………………(1)
Difference between 1000th terms of these A.P.s
(a1 + 999d) − (a2 + 999d) = a1 − a2
From equation (1),
This difference, a1 − a2 = 100
Hence, the difference between 1000th terms of these A.P. will be 100.
13. How many three digit numbers are divisible by 7?
Ans: three-digit number that is divisible by 7 = 105
Next number = 105 + 7 = 112
Therefore, 105, 112, 119, …
All are three digit numbers which are divisible by 7 and thus, all these
are terms of an A.P. having first term as 105 and common difference as 7.
The maximum possible three-digit number is 999. When we divide it by
7, the remainder will be 5. Clearly, 999 − 5 = 994 is the
maximum possible three-digit number that is divisible by 7.
The series is as follows.
105, 112, 119, …, 994
Let 994 be the nth term of this A.P.
a = 105, d = 7 and an = 994. n = ?
an = a + (n − 1) d
994 = 105 + (n − 1) 7
889 = (n − 1) 7
(n − 1) = 127
n = 128
Therefore, 128 three-digit numbers are divisible by 7.
14. How many multiples of 4 lie between 10 and 250?
Ans: First multiple of 4 that is greater than 10 is 12. Next will be 16. Therefore, 12, 16, 20, 24, …
All these are divisible by 4 and thus, all these are terms of an
A.P. with
first term as 12 and common difference as 4.
When we divide 250 by 4, the remainder will be 2.
Therefore, 250 − 2 = 248 is divisible by 4.
The series is as follows.
12, 16, 20, 24, …, 248
Let 248 be the n
th term of this A.P.
15. For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10,17, … equal .
Ans: For AP: 63, 65, 67, …
a = 63 and d = a2 − a1 = 65 − 63 = 2
nth term of this A.P. = an = a + (n − 1) d
an= 63 + (n − 1) 2 = 63 + 2n − 2
an = 61 + 2n ……………(1)
For AP:3, 10, 17, …
a = 3 and d = a2 − a1 = 10 − 3 = 7
nth term of this A.P. = 3 + (n − 1) 7
an = 3 + 7n − 7
an = 7n − 4 …………………………………(2)
It is given that, nth term of these A.P.s are equal to each other.
Equating both these equations, we obtain
61 + 2n = 7n − 4
61 + 4 = 5n
5n = 65
n = 13
Therefore, 13th terms of both these A.P.s are equal to each other.
16: Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12.
Ans: a3 = 16
a + (3 – 1) d = 16
a + 2d = 16 ………………………….(1)
a7 − a5 = 12
[a+ (7 − 1) d] − [a + (5 − 1) d]= 12
(a + 6d) − (a + 4d) = 12
2d = 12
d = 6
From equation (1), we obtain
a + 2 (6) = 16
a + 12 = 16
a = 4
Therefore, A.P. will be 4, 10, 16, 22, …
17. Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253
Ans: Given A.P. is 3, 8, 13, …, 253
Common difference for this A.P. is 5.
Therefore, this A.P. can be written in reverse order as
253, 248, 243, …, 13, 8, 5
For this A.P.,
a = 253
d = 248 − 253 = −5
n = 20
a20 = a + (20 − 1) d
a20 = 253 + (19) (−5)
a20 = 253 − 95 a = 158
Therefore, 20
20th term from the last term is 158.
18. The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the A.P.
Ans: We know that, an = a + (n − 1) d
a4 = a + (4 − 1) d
a4 = a + 3d
Similarly,
a8 = a + 7d
a6 = a + 5d
a10 = a + 9d
Given that, a4 + a8 = 24
a + 3d + a + 7d = 24
2a + 10d = 24
a + 5d = 12 …………………………….(1)
a6 + a10 = 44
a + 5d + a + 9d = 44
2a + 14d = 44
a + 7d = 22 …………………………….(2)
On subtracting equation (1) from (2), we obtain
2d = 22 − 12
2d = 10
d = 5
From equation (1), we obtain
a + 5d = 12
a + 5 (5) = 12
a + 25 = 12
a = −13
a2 = a + d = − 13 + 5 = −8
a3 = a2 + d = − 8 + 5 = −3
Therefore, the first three terms of this A.P. are −13, −8, and −3.
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?
Ans: It can be observed that the incomes that Subba Rao obtained in various years are in A.P. as every year, his salary is increased by Rs 200.
Therefore, the salaries of each year after 1995 are
5000, 5200, 5400, …
Here, a = 5000 and d = 200
Let after nth
th year, his salary be Rs 7000.
Therefore, an = a + (n − 1) d
7000 = 5000 + (n − 1) 200
200(n − 1) = 2000
(n − 1) = 10
n = 11
Therefore, in 11th year, his salary will be Rs 7000.
20. Ramkali saved 𝑅𝑠 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her week, her weekly savings become Rs 20.75, find n.
Ans: aₙ = a + (n – 1)d is the nth term of AP.
Here, a is the first term, d is a common difference and n is the number of terms.
So, in the 10th week Ramkali’s saving will be ₹ 20.75.
| Exercise 5.3 |
1. Find the sum of the following APs.
(i) 2, 7, 12 ,…., to 10 terms.
Ans: 2, 7, 12 ,…, to 10 terms
For this A.P., a = 2, d = a2 − a1
= 7 − 2 = 5 and n = 10
We know that,
(ii) − 37, − 33, − 29 ,…, to 12 terms
Ans: For this A.P., a = −37, d = a2 − a1
= (−33) − (−37) = − 33 + 37 = 4
n = 12
We know that,
(iii) 0.6, 1.7, 2.8 ,…….., to 100 terms
Ans: 0.6, 1.7, 2.8 ,…, to 100 terms For this A.P., a = 0.6, d = a2 − a1 = 1.7 − 0.6 = 1.1 and n = 100
We know that,
2. Find the sums given below
(i) 7 + 10 1/2+ 14 + ………… + 84
Ans:
(ii) 34 + 32 + 30 + ……….. + 10
Ans: For this A.P., a = 34, d = a2 − a1 = 32 − 34 = −2 and l = 10
Let 10 be the n
th term of this A.P. l = a + (n − 1) d
10 = 34 + (n − 1) (−2)
−24 = (n − 1) (−2)
12 = n − 1
n = 13
(iii) − 5 + (− 8) + (− 11) + ………… + (− 230)
Ans: For this A.P.,
a = −5, l = −230 and d = a2 − a1 = (−8) − (−5) = − 8 + 5 = −3
Let −230 be the n
th term of this A.P.
l = a + (n − 1)d
−230 = − 5 + (n − 1) (−3)
−225 = (n − 1) (−3)
(n − 1) = 75
n = 76
3. In an AP
(i) Given a = 5, d = 3, an = 50, find n and Sn.
Ans: Given that, a = 5, d = 3, an = 50
As an = a + (n − 1)d,
∴ 50 = 5 + (n − 1)3
45 = (n − 1)3
15 = n − 1 n = 16
(ii) Give a13 = 7, d = 35 , find d and s13
Ans: Given that, a = 7, a13 = 35
As an = a + (n − 1) d,
∴ a13 = a + (13 − 1) d
35 = 7 + 12 d
35 − 7 = 12d
28 = 12d
(iii) Give a12 = 37, d = 3 find a and s12
Ans: Given that, a, = 15, S10 = 125
As an = a + (n − 1)d,
a3 = a + (3 − 1)d
15 = a + 2d ……………………….(i)
On multiplying equation (i) by 2, we obtain
30 = 2a + 4d …………………..(iii)
On subtracting equation (iii) from (ii), we obtain
−5 = 5d
d = −1
From equation (i),
15 = a + 2(−1)
15 = a − 2
a = 17
a10 = a + (10 − 1)d
a10 = 17 + (9) (−1)
a10 = 17 − 9 = 8
Ans: As an = a + (n − 1)d,
a12 = a + (12 − 1)3
37 = a + 3
(v) Given d = 5, S9 = 75, find a and a9.
Ans:
(vi) Given a = 2, d = 8, Sn = 90, find n and an.
Ans: As an = a + (n − 1)d,
a3 = a + (3 − 1)d
15 = a + 2d ……………………….(i)
90 = n (4n − 2) = 4n 2 − 2n
4n2 − 2n − 90 = 0
4n2 − 20n + 18n − 90 = 0
4n (n − 5) + 18 (n − 5) = 0
(n − 5) (4n + 18) = 0
Either n − 5 = 0 or 4n + 18 = 0
n = 5 or
However, n can neither be negative nor fractional.
Therefore, n = 5
an = a + (n − 1)d
a, = 2 + (5 − 1)8
= 2 + (4) (8)
= 2 + 32 = 34
(vii) Given a = 8, an = 62, Sn = 210, find n and d.
Ans:
(viii) Given an = 4, d = 2, Sn = − 14, find n and a.
Ans: Given that, an = 4, d = 2, Sn = −14
an = a + (n − 1)d
4 = a + (n − 1)2
4 = a + 2n − 2
a + 2n = 6
a = 6 − 2n …………………………….(i)
−28 = n (a + 4)
−28 = n (6 − 2n + 4) {From equation (i)}
−28 = n (− 2n + 10)
−28 = − 2n 2 + 10n
2n2 − 10n − 28 = 0
n2 − 5n −14 = 0
n2 − 7n + 2n − 14 = 0
n (n − 7) + 2(n − 7) = 0
(n − 7) (n + 2) = 0
Either n − 7 = 0 or n + 2 = 0
n = 7 or n = −2
However, n can neither be negative nor fractional.
Therefore, n = 7
From equation (i), we obtain
a = 6 − 2n
a = 6 − 2(7)
= 6 − 14
= −8
(ix) Given a = 3, n = 8, S = 192, find d.
Ans:
192 = 4 [6 + 7d]
48 = 6 + 7d
42 = 7d
d = 6
(x) Given l = 28, S = 144 and there are total 9 terms. Find a.
Ans:
(16) × (2) = a + 28
32 = a + 28
a = 4
4. How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?
Ans: Let there be n terms of this A.P. For this A.P.,
a = 9 and d = a2 − a1 = 17 − 9 = 8
636 = n [9 + 4n − 4]
636 = n (4n + 5)
4n2 + 5n − 636 = 0
4n2 + 53n − 48n − 636 = 0
n (4n + 53) − 12 (4n + 53) = 0
(4n + 53) (n − 12) = 0
Either 4n + 53 = 0 or n − 12 = 0
n cannot be -53/4, As the number of terms can neither be negative nor fractional, therefore, n = 12 only.
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.
Ans: Given that, a = 5, l = 45 and Sn = 400
n = 16 l = a + (n− 1) d
45 = 5 + (16 − 1) d
40 = 15d
6. 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?
Ans: Given that, a = 17, l = 350 and d = 9
Let there be n terms in the A.P.
l = a + (n − 1) d
350 = 17 + (n − 1)9
333 = (n − 1)9
(n − 1) = 37
n = 38
Thus, this A.P. contains 38 terms and the sum of the terms of
this A.P. is 6973.
7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
Ans: d = 7 and a22 = 149. S22 =?
an = a + (n − 1)d
a22 = a + (22 − 1)d
149 = a + 21 × 7
149 = a + 147
a = 2
8. Find the sum of the first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
Ans: Given that, a2 = 14 and a3 = 18
d = a3 − a2 = 18 − 14 = 4
a2 = a + d
14 = a + 4
a = 10
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.
Ans: Given that, S7 = 49
S17 = 289
17 = (a + 8d)
a + 8d = 17 ……………………….(ii)
Subtracting equation (i) from equation (ii),
5d = 10
d = 2
From equation (i),
a + 3(2) = 7
a + 6 = 7
a = 1
10. Show that a1, a2 … , an , … form an AP where an is defined as below:
(i) an = 3 + 4n
Ans: an = 3 + 4n
a1 = 3 + 4(1) = 7
a2 = 3 + 4(2) = 3 + 8 = 11
a3 = 3 + 4(3) = 3 + 12 = 15
a4 = 3 + 4(4) = 3 + 16 = 19
It can be observed that
a2 − a1 = 11 − 7 = 4
a3 − a2 = 15 − 11 = 4
a4 − a3 = 19 − 15 = 4
i.e., ak + 1 − ak is the same every time. Therefore, this is an AP
with a common difference as 4 and first term as 7.
11. If the sum of the first n terms of an AP is 4n − n 2 , what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly find the 3rd, the10th and the n th terms.
Ans: Given that,
Sn = 4n − n2
First term, a = S1 = 4(1) − (1)2 = 4 − 1 = 3
Sum of first two terms = S2 = 4(2) − (2)2 = 8 − 4 = 4 Second
term, a2 = S2 − S1 = 4 − 3 = 1
d = a2 − a = 1 − 3 = −2
an = a + (n − 1)d
= 3 + (n − 1) (−2)
= 3 − 2n + 2
= 5 − 2n
Therefore,
a3 = 5 − 2(3) = 5 − 6 = −1
a10 = 5 − 2(10) = 5 − 20 = −15
Hence, the sum of the first two terms is 4. The second term is 1.
3rd, 10th and nth terms are −1, −15, and 5 − 2n respectively.
12. Find the sum of the first 40 positive integers divisible by 6.
Ans: The positive integers that are divisible by 6 are
6, 12, 18, 24 …
It can be observed that these are making an A.P. whose first term is 6 and common difference is 6.
a = 6 and d = 6
S40 =?
= 20[12 + (39) (6)]
= 20(12 + 234)
= 20 × 246
= 4920
13. Find the sum of the first 15 multiples of 8.
Ans: The multiples of 8 are
8, 16, 24, 32…
These are in an A.P., having first term as 8 and common difference as 8.
Therefore, a = 8 and d = 8
S15 =?
14. Find the sum of the odd numbers between 0 and 50.
Ans: The odd numbers between 0 and 50 are 1, 3, 5, 7, 9 … 49
Therefore, it can be observed that these odd numbers are in an A.P.
a = 1, d = 2 and l = 49
l = a + (n − 1) d
49 = 1 + (n − 1)2
48 = 2(n − 1)
n − 1 = 24
n = 25
15. 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.
Ans: It can be observed that these penalties are in an A.P.
having first term
as 200 and common difference as 50.
a = 200 and d = 50
Penalty that has to be paid if he has delayed the work by 30 days = S30
= 15 [400 + 1450]
= 15 (1850)
= 27750
Therefore, the contractor has to pay Rs 27750 as penalty.
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.
Ans: Let the cost of 1st prize be P.
Cost of 2nd prize = P − 20
And cost of 3rd prize = P − 40
It can be observed that the cost of these prizes are in an A.P. having
common difference as −20 and first term as P.
a = P and d = −20
Given that, S7 = 700
a + 3(−20) = 100
a − 60 = 100
a = 160
Therefore, the value of each of the prizes was Rs 160, Rs 140,
Rs 120,
Rs 100, Rs 80, Rs 60, and Rs 40.
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 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?
Ans: It can be observed that the number of trees planted by the students is in
an AP. 1, 2, 3, 4, 5………………..12
First term, a = 1
Common difference, d = 2 − 1 = 1
Therefore, number of trees planted by 1 section of the classes = 78
Number of trees planted by 3 sections of the classes = 3 × 78 = 234
Therefore, 234 trees will be planted by the students.
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, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semi-circles?
Ans: Semi-perimeter of circle = 𝜋𝑟
I1 = 𝜋 (0.5) = 𝜋/2 cm
I2 = 𝜋 (1) = 𝜋 𝑐𝑚
I3 = 𝜋 (1.5) = 3𝜋/2𝑐𝑚
Therefore, I1, I2, I3 ,i.e. the lengths of the semi-circles are in an A.P.,
We know that the sum of n terms of an a A.P. is given by
Therefore, the length of such spiral of thirteen consecutive semi-circles will be 143 cm.
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. In how many rows are the 200 logs placed and how many logs are in the top row?
Ans: It can be observed that the numbers of logs in rows are in an A.P.
20, 19, 18…For this A.P.,
a = 20 and d = a2 − a1 = 19 − 20 = −1
Let a total of 200 logs be placed in n rows.
Sn = 200
400 = n (40 − n + 1)
400 = n (41 − n)
400 = 41n − n2
n2 − 41n + 400 = 0
n2 − 16n − 25n + 400 = 0
n (n − 16) −25 (n − 16) = 0
(n − 16) (n − 25) = 0
Either (n − 16) = 0 or n − 25 = 0
n = 16 or n = 25
an = a + (n − 1)d
a16 = 20 + (16 − 1) (−1)
a16 = 20 − 15
a16 = 5
Similarly, a25 = 20 + (25 − 1) (−1)
a25 = 20 − 24
= −4
20. 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 in a straight line. There are ten potatoes in the line.
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 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 the second potato, the total distance (in metres) run by a competitor is 2 × 5 + 2 ×(5 + 3)]
Ans:
The distances of potatoes are as follows.
5, 8, 11, 14…
It can be observed that these distances are in A.P.
a = 5 d = 8 − 5 = 3
As every time she has to run back to the bucket, therefore, the total distance that the competitor has to run will be two times of it. Therefore, total distance that the competitor will run = 2 × 185 = 370 m
Alternatively,
The distances of potatoes from the bucket are 5, 8, 11, 14… Distance run
by the competitor for collecting these potatoes are two times of the distance at which the potatoes have been kept.
Therefore, distances to be run are 10, 16, 22, 28, 34,……….
a = 10 and d = 16 − 10 = 6. S10 =?
Therefore, the competitor will run a total distance of 370 m.
| Exercise 5.4 |
1. Which term of the A.P. 121, 117, 113 … is its first negative term?
Given A.P. is 121, 117, 113 …
a = 121 d = 117 − 121 = −4
an = a + (n − 1) d
= 121 + (n − 1) (−4)
= 121 − 4n + 4
= 125 − 4n
We have to find the first negative term of this A.P.
The sum of the third and the seventh terms of an A.P is 6 and their product is 8. Find the sum of the first sixteen terms of the A.P.
2. The sum of the third and the seventh terms of an A.P is 6 and their product is 8. Find the sum of first sixteen terms of the A.P.
Ans: We know that,
an = a + (n − 1) d
a3 = a + (3 − 1) d
a3 = a + 2d
Similarly, a7 = a + 6d
Given that,
a3 + a7 = 6
(a + 2d) + (a + 6d) = 6
2a + 8d = 6
a + 4d = 3
a = 3 − 4d ……………………(i)
Also,
it is given that (a3) × (a7) = 8
(a + 2d) × (a + 6d) = 8
From equation (i),
3. A ladder has rungs 25 cm apart. (See figure). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and bottom rungs are 2 1 2 m apart, what is the length of the wood required for the rungs?
Ans: It is given that the rungs are 25 cm apart and the top and bottom rungs
Now, as the lengths of the rungs decrease uniformly, they will be in an A.P. The length of the wood required for the rungs equals the sum of all the terms of this A.P. First term, a = 45 Last term, l = 25 and n = 11
Therefore, the length of the wood required for the rungs is 385 cm.
4. The houses of a row are number consecutively from 1 to 49. Show that there is a value of x such that the sum of numbers of the houses preceding the house numbered x is equal to the sum of the number of houses following it. Find this value of x.
Ans: The number of houses was
1, 2, 3 … 49
It can be observed that the number of houses are in an A.P.
having a as1 and d also as 1.
Let us assume that the number of x
the house was like this.
We know that,
Sum of n terms in an A.P. =𝑛2
[2𝑎 + (𝑛 − 1)𝑑]
Sum of number of houses preceding x
th house = Sx − 1
Sum of number of houses following xth house = S49 − Sx
However, the house numbers are positive integers. The value of x will be 35 only. Therefore, house number 35 is such that the sum of the numbers of houses preceding the house numbered 35 is equal to the sum of the numbers of the houses following it.
5. 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 ¼ m and a tread of ½ m (See figure) calculate the total volume of concrete required to build the terrace.
Ans:
From the figure, it can be observed that
the 1st step is ½ m wide, 2nd step is 1 m wide, 3rd step is 3/2 m wide. Therefore, the width of each step is increasing by ½ m each time Whereas their height ¼ m and length 50 m remains the same. Therefore, the widths of these steps are
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