Arithmetic Progression Questions and Answers

Arithmetic Progression Questions and Answers for Practice

There are a number of Arithmetic progression Questions and they have varied formulas for solving the sequence.  In this page Arithmetic Progression Questions and Answers are given.
Arithmetic Progressions Questions and Answers

Notations in Arithmetic Progression (AP)

  • Initial term (a): In an arithmetic progression, the first number in the series is called the initial term.
     
  • Common difference (d):  The ‘d’ denotes the variance between all the consecutive terms of the progression. 
     
  • Nth term (a_{n}): The a_{n} defines the terms of the progression or sequence whereas ‘n denotes the position of the given number or alphabet which can be obtained by a_{n} = a + (n − 1)d.
     
  • Sum of AP (Sn): The sum of an AP can be obtained by Sn = \frac{n}{2} [2a + (n − 1) d]

 If ‘n’ is equal to one then it denotes the initial term of arithmetic progression. This is known as the precise method for an Arithmetic progression.

General form of AP:

The general form of an AP is : a, a + d, a + 2d, a + 3d, …….

Arithmetic Progression Questions and Answers

Finite or Infinite Arithmetic Progressions

  • Finite Arithmetic Progression

When there are limited number of terms in the sequence then it is known as Finite Arithmetic Progression.

For example: 10, 20, 30, 40, 50

  • Infinite Arithmetic Progression

When there are unlimited number of terms in the sequence then it is known as Infinite Arithmetic Progression.

For example: 3, 5, 7, 9, 11, 13, ..…….

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Arithmetic Progression Questions and Answers 

1. Find the 1st term of the AP whose 6th and 10th terms are respectively 36 and 56.

8

8

15.03%

9

9

12.42%

10

10

3.92%

11

11

68.63%

As we know about the nth term of AP is [a + (n-1)d]
So, 6th term = a + 5d
and 10th term = a + 9d
Given a + 5d = 36 ………(1)
a + 9d = 56……….(2)
Subtract (1) form (2)
4d = 20
d = 5
put value of d in (1)
a + 5×5 = 36
a =11
Hence, 1st term of the AP is 11.

2. A figure 24 is distributed into three parts which are in Arithmetic Progression and total of their squares is 208.

8.5

8.5

21.24%

9

9

11.5%

10.9

10.9

53.98%

11

11

13.27%

Let the three consecutive parts of AP are (a-d), a, (a+d).
Given that
(a-d) + a + (a+d) = 24
3a = 24
a = 8
Again, (a-d)2 + a2 + (a+d)2 = 208
a2 + d2 – 2ad + a2 + a2 + d2 + 2ad = 208
3a2 + 2d2 = 208
put value of a
3(8)2 + 2d2 = 208
2d2 = 208 – 192
d2 = 8
d = ∓2.9
Hence, the largest part is (a+d) = 8+2.9 = 10.9

3. How many natural numbers between 300 to 500 are multiples of 40?

3

3

4.8%

4

4

14.4%

5

5

71.2%

6

6

9.6%

The series have multiple starting from 320, 360, ……..480
It becomes an AP having common difference 40.
Total number of natural numbers =

n=  [ \frac{(l-a)}{d} ]+ 1

= [\frac{480 – 320}{40}] + 1
= \frac{160}{40} + 1 = 5

4. In the given arithmetic progression, ’33’ would be a term in it. 7, 10, 13, 16, 19, 22………52.

True

True

13.49%

False

False

72.22%

can  not say

can  not say

7.94%

None

None

6.35%

nth term = a + (n-1)d, 33 = 7 + (n-1)3,

n = \frac{29}{3}

n = 9.666

Clearly 9.66 is not an integer. So 33 is not a term in this series.

5. In the given arithmetic progression, ’44’ would be a term in it.
17, 20, 23, 26, 29, 32………62.

True

True

82.05%

False

False

16.24%

can not say

can not say

0.85%

None

None

0.85%

nth term = a + (n-1)d

44 = 17 + (n-1)3,

n = \frac{30}{3} = 10

n = 10

Clearly 10 is an integer. So 44 is a term in this series.

6. An elastic toy bounces (\frac{3}{4})th of its height after touching to the base from which it has fall over. Calculate the full distance that it travels before coming to rest, if it is mildly fallen from a top of 360 metres.

5400

5400

14.49%

5300

5300

11.59%

2520

2520

60.87%

4800

4800

13.04%

It becomes an infinite sum of series.
So, use \frac{a}{(1-r)} to calculate the distance
Ball rebounds to \frac{3}{4} of its height -> 360\times (\frac{3}{4}) = 270
360 + \frac{270 \times 2}{1-(\frac{3}{4})}
= 360 + 2160
= 2520
Hence, the full distance of elastic toy=2520

7. Anupam joins a company Pidilite industries in January 2018 and he get his first pay of Rs 2000. After that he got an increment every month of Rs 1500. Calculate his total pay after the end of 5 years of his job.

80000

80000

16.28%

85000

85000

4.65%

90500

90500

59.3%

92000

92000

19.77%

It is an AP 2000, 3500, 5000, ……..so on.
As we know in total there 60 months in 5 years.
We need to calculate the 60th term of the series.
Common difference d = 1500
a60 = a + (n-1)d
a60 = 2000 + 59 x 1500
a60 = 2000 + 88500 = 90500
After completion of 5 years of service his salary will be Rs 90500.

8. The sequence 9√5, 8√5, 7√5, …….. is

Arithmetic progression

Arithmetic progression

78.18%

Geometric progression

Geometric progression

8.18%

Harmonic progression

Harmonic progression

9.09%

None of these

None of these

4.55%

Difference is:

d= 8√5-9√5 = 7√5-8√5 = -√5

Therefore given arrangement is in Arithmetic progression.

9. 8th term of the series 2√4+√4+0+..... will be

−5√4

−5√4

77.89%

4√4

4√4

9.47%

10√2

10√2

4.21%

-10√2

-10√2

8.42%

Clearly the given series is in Arithmetic progression

2√4+√4+0+.....  is an A.P.

Now

a=2√4, d=−√4

. Hence 8th term of the series

=2√4+(8−1)(−√4)=−5√4.

10. If the 8th term of an Arithmetic Progression be zero, then the ratio of its  28th  and 18th  term will be?

1 : 2

1 : 2

8.89%

2 : 2

2 : 2

7.78%

2 : 1

2 : 1

75.56%

3 : 1

3 : 1

7.78%

Given that 8th term = 0

8th term = a+(8−1)d = a+7d

so a+7d = 0

Now ratio of 28th and 18th terms

= \frac{a+27d}{a+17d} = \frac{(a+7d)+20d}{(a+7d)+10d}=\frac{20d}{10d} = \frac{2}{1} = 2:1

11. Which among the below is an arithmetic sequence

f(n)=an+b;n∈N

f(n)=an+b;n∈N

61.36%

f(n)=krn;n∈N

f(n)=krn;n∈N

12.5%

f(n)=(an+b)krn;n∈N

f(n)=(an+b)krn;n∈N

15.91%

f(n)=1a(n+bn);n∈N

f(n)=1a(n+bn);n∈N

10.23%

Sequence

f(n)=an+b; n∈N is an A.P. Putting n=1, 2, 3, 4, ..........,

we get the sequence (a+b), (2a+b), (3a+b),.........

which  is  an  A.P. Where first term (A)=(a+b) and common difference;d=a

12. Which term of the sequence

(−10+19i),(−8+16i), (−6+13i),......is unreal

4th

4th

11.54%

5th

5th

10.26%

6th

6th

64.1%

7th

7th

14.1%

Given sequence is

(−10+19i)(−8+16i)(−6+13i)(−4+10i)(−2+7i)(−0+4i)

Therefore 6th term is purely imaginary.

13. The nth term of the following series (1×3)+(3×5)+(5×7)+(7×9)+....... will be

n(2n+1)

n(2n+1)

14.46%

2n(2n-1)

2n(2n-1)

8.43%

4n+1

4n+1

7.23%

(2n-1)(2n+1)

(2n-1)(2n+1)

69.88%

Since, given series is (1×3)+(3×5)+(5×7)+(7×9)+.......

In 1 , 3 , 5.....

nth term = 1+(n-1)2 = (2n-1)

In 3 , 5 , 7.....

nth term = 3+(n-1)2 = (2n+1)

Hence nth term of the series is (2n-1)(2n+1).

14. The number of terms in the series 201+199+197+.....+147 is

25

25

16.25%

28

28

77.5%

30

30

5%

20

20

1.25%

Given series 201+199+197+.........+147

So, first term a=201, difference d = −2 and last term, l = 147

As per the formula

T_{l}=a+(n−1)d

=>147=201+(n−1)(−2)

=>−54=(n−1)(−2)

n=28

15. Calculate the 17th term of Arithmetic Progression whose first term is 3 and common difference is 4.

60

60

2.53%

65

65

6.33%

70

70

2.53%

67

67

88.61%

nth term of A.P. =a+(n-1)d = 3+(17-1)4 = 67.

16. The sum of first 200 numbers, 1 to 200 is divisible by?

2

2

5.06%

2,4 and 5

2,4 and 5

75.95%

1,2,4,8

1,2,4,8

12.66%

None of the above

None of the above

6.33%

as per the formula:

(n\times \frac{(n+1)}{2})

\frac{(200\times 201)}{2}

= 100 x 201

201 is an odd number and 100 is divisible by 2.

So 100 x 201 will be divisible by 2,4 and 5.

17. How many numbers between 10 and 89 divisible by 8?

8

8

5.13%

9

9

8.97%

10

10

76.92%

11

11

8.97%

Clearly the numbers which will be divisible by 8:

16, 24,32, ………………. 88

This is an Arithmetic Progression where a = 16 and common difference = 8

Thus: T_{n} = 88

or a+(n-1)d=88

or 16+(n-1)8=88

or 16+8n-8=88;

8n=80

n = 10

18. The sum of 3 numbers in arithmetic progression is 36 and product of their extreme is 80. Find the numbers.

20,8,16

20,8,16

3.95%

4, 12, 20

4, 12, 20

80.26%

12, 20, 28

12, 20, 28

9.21%

None of the above

None of the above

6.58%

Assume the numbers as:

a-d, a, a+d

or a-d+a+a+d=36

or 3a=36

a=12

now (a-d) (a+d)=80

a²-d²=80

144-d²=80

d²=64

d=8

thus the numbers will be: 4, 12 and 20

19. Find the 11th term of the AP : 3, 11, 19, . . .

78

78

4%

80

80

4%

83

83

88%

76

76

4%

Here, a = 3, d = 11 – 3 = 8 and n = 11.

We have a_{n} = a + (n – 1) d

= 3 + (11 – 1) × 8 = 3 + 80 = 83 Therefore, the 11th term of the given AP is 83.

20. Find the 10th term from the last term (towards the first term) of the AP : 11, 7, 3, . . ., – 65.

29

29

10.67%

-29

-29

77.33%

32

32

2.67%

-32

-32

9.33%

Here, a = 11, d = 7 – 11 = – 4, l = – 65, where l = a + (n – 1) d

but as per the information given in the question If we write the given AP in the reverse order, then a = – 65 and d = 4. So, we have to find the 10th term with new a and d.

so = – 65 + (10 – 1) × 4 = – 65+ 36 = – 29

So, the 10th term will be – 29.

 

21. In a vegetable market, there are 25 tulip plants in the first row, 23 in the second, 21 in the third, and so on. There are 5 tulip plants in the last row. Calculate the total number of rows?

6

6

5.41%

7

7

2.7%

11

11

85.14%

13

13

6.76%

Clearly the above mentioned series forms an AP series.

Assume the total number of rows as n.

Then a = 25, d = 23 – 25 = – 2,

a_{n} = a + (n – 1) d We have, 5 = 25 + (n – 1)(– 2)

i.e., – 20 = (n – 1)(– 2) i.e., n = 11 So, there are total 11 rows.

22. Find the sum of first 27 terms of the list of numbers whose nth term is given by

An=3+2n

820

820

4.62%

827

827

9.23%

837

837

80%

849

849

6.15%

The nth term is an=3+2n

so by using this we can find the value of terms like a_{1} = 5,a_{2}  = 7,a_{3} = 9

and a_{27}=57

now the first term is 5 and the last term is 57 and the common difference is 2

By applying the formula we will get:

S_{n} = \frac{n}{2}(2a+(n-1)d)

so S_{27} = \frac{27}{2}(10+26\times 2)

= 837

23. In the following AP series calculate the number of terms
15, 18, 21, 24, 27, 30………60.

13

13

6.76%

11

11

5.41%

16

16

77.03%

15

15

10.81%

nth term = first term + (number of terms – 1) common difference.

60 = 15 + (n-1)3,

60=15+3n-3=

48=3n

n = 16

24. In the following AP series the term at position 12 will be?
4, 8, 12, 16, 20………60.

54

54

6.67%

48

48

89.33%

45

45

1.33%

None of the mentioned.

None of the mentioned.

2.67%

nth term = a + (n – 1)d,

nth term = 4+(12-1)4

= 4+44

= 48

25. In the following arithmetic progression calculate the position of first negative term?
150, 147, 144, 141,…………

57

57

4.23%

52

52

59.15%

58

58

4.23%

51

51

32.39%

d = 147 - 150 = -3

Assume nth term=0 , the following term will be first negative term.
0 = 150 + (n-1) – 3,

0 = 150-3n+3

3n=153

n = 51

So at n = 51 + 1 =52 the first negative term would occur.

26. If n is an integer then which among the below sequences in AP will have common difference 3?

A_{n} = 2n² + 3n

A_{n} = 2n² + 3n

5.56%

A_{n} = 2n² + 3

A_{n} = 2n² + 3

16.67%

A_{n} = 5 + 3n

A_{n} = 5 + 3n

75%

A_{n} = 3n² + 3n

A_{n} = 3n² + 3n

2.78%

A_{n} = 5 + 3n is a linear expression. This contains AP with difference of 3.

27. If x, y, and z are in AP then relation between x, y, z can be

2y = 2x + 3z

2y = 2x + 3z

8.11%

2x = y + z

2x = y + z

21.62%

2y = x + z

2y = x + z

58.11%

2z = x + z

2z = x + z

12.16%

The term y should be the arithmetic mean of term x and z.

28. If the total of 3 successive terms in AP is 180 then mid of those three terms will be:

50

50

2.67%

60

60

85.33%

70

70

8%

80

80

4%

Let a-d , a ,a+d three terms in AP

Sum = 180

a-d+a+a+d = 180

a = 60

Mid term = a = 60

29. In the given arithmetic progression, ’26’ would be a term in it.
6, 9, 12, 15, 18, 21………51.

True

True

16.67%

False

False

76.92%

can not say

can not say

1.28%

None

None

5.13%

nth term = a + (n-1)d,

26 = 6 + (n-1)3,

n = \frac{23}{3}

n = 7.666

Clearly 7.66 is not an integer. So 26 is not a term in this series.

30. Which among the below is correctly representing an AP series?

1, 3, 5, 7, 9

1, 3, 5, 7, 9

69.14%

2, 4, 8, 16, 32

2, 4, 8, 16, 32

13.58%

3, 9, 27, 81, 243

3, 9, 27, 81, 243

8.64%

1, 2, 4, 7, 10, 15

1, 2, 4, 7, 10, 15

8.64%

In A.P. the difference in any term with previous term is same.

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