Exercise 5.1 (NCERT) – Arithmetic Progressions, Class 10
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| Chapter 5 | Exercise 5.1 | Class 10 |
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Below are the solutions of every question of exercise 5.1 of chapter 5, Arithmetic Progressions, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.
Question 1
In which of the following situations, does the list of numbers involved 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.
Fare for
1 km = ₹15
2 km = 15 + 8 = ₹23
3 km = 15 + 8 + 8 = ₹31
4 km = 15 + 8 + 8 + 8 = ₹39
and so on…
We get the list:
15, 23, 31, 39, …
This is an AP series because the difference between successive terms is constant = 8.
Answer: Yes, It is an AP series
(ii) 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.
Considering the original amount of air (a1)= 1 Unit
After the first removal, the remaining amount of air is:
After the second removal, the remaining amount of air is:
Let us find the common difference
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: No, It is not an AP series
(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.
Digging cost
For 1 metre: ₹150
For 2 metres: 150 + 50 = ₹200
For 3 metres: 150 + 50 + 50 = ₹250
For 4 metres: 150 + 50 + 50 + 50 = ₹300
and so on…
We get the list:
150, 200, 250, 300, …
This is an AP series because the difference between successive terms is constant = 50.
Answer: Yes, It is an AP series
(iv) The amount of money in the account every year, when Rs. 10000 is deposited at compound interest at 8 % per annum.
Amount in the account
For the 1st year:
For the 2nd year:
For the 3rd year:
We get the list:
10000, 10800, 11664, 12597, …
We can see that
A2 – A1 = 11664 – 10800 = 864
A3 – A2 = 12597 – 11664 = 933
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: No, It is not an AP series
Question 2
Write first four terms of the AP, when the first term a and the common difference d are given as follows:
(i) a = 10, d = 10
First term
Common difference
First four terms are: a, a + d, a + 2d, a + 3d
a₁ = 10
a₂ = 10 + 10 = 20
a₃ = 10 + 20 = 30
a₄ = 10 + 30 = 40
Answer: 10, 20, 30, 40
(ii) a = –2, d = 0
First term
Common difference
First four terms are: a, a + d, a + 2d, a + 3d
a₁ = –2
a₂ = –2 + 0 = –2
a₃ = –2 + 0 = –2
a₄ = –2 + 0 = –2
Answer: -2, -2, -2, -2
(iii) a = 4, d = – 3
First term
Common difference
First four terms are: a, a + d, a + 2d, a + 3d
a₁ = 4
a₂ = 4 + (–3) = 1
a₃ = 1 + (–3) = –2
a₄ = –2 + (–3) = –5
Answer: 4, 1, –2, –5
(iv) a = – 1, d = 1/2
First four terms are: a, a + d, a + 2d, a + 3d
Answer: –1, –1/2, 0, 1/2
(v) a = – 1.25, d = – 0.25
First term
Common difference
First four terms are: a, a + d, a + 2d, a + 3d
a₁ = –1.25
a₂ = –1.25 + (–0.25) = –1.50
a₃ = –1.50 + (–0.25) = –1.75
a₄ = –1.75 + (–0.25) = –2.00
Answer: –1.25, –1.50, –1.75, –2.00
Question 3
For the following APs, write the first term and the common difference:
(i) 3, 1, – 1, – 3, . . .
First term (a) = 3
Common difference (d) = a₂ – a₁ = 1 – 3 = –2
(ii) – 5, – 1, 3, 7, . . .
First term (a) = –5
Common difference (d) = a₂ – a₁ = (–1) – (–5) = –1 + 5 = 4
(iii) 1/3, 5/3, 9/3, 13/3
(iv) 0.6, 1.7, 2.8, 3.9, . . .
First term (a) = 0.6
Common difference (d) = a₂ – a₁ = 1.7 – 0.6 = 1.1
Question 4
Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
(i) 2, 4, 8, 16, . . .
Given sequence
2, 4, 8, 16, …
Checking the differences
a₂ − a₁ = 4 − 2 = 2
a₃ − a₂ = 8 − 4 = 4
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: Not an AP
(iii) 2, 5/2, 3, 7/2, …
Given sequence
2, 2.5, 3, 3.5, …
Checking the differences
a₂ – a₁ = 2.5 – 2 = 0.5
a₃ – a₂ = 3 – 2.5 = 0.5
a₄ – a₃ = 3.5 – 3 = 0.5
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = 0.5
Three more terms
a₅ = 3.5 + 0.5 = 4.0
a₆ = 4.0 + 0.5 = 4.5
a₇ = 4.5 + 0.5 = 5.0
Answer: It is an AP series. Next three terms are 4, 4.5, 5
(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . .
Given sequence
–1.2, –3.2, –5.2, –7.2, …
Checking the differences
a₂ − a₁ = –3.2 − (–1.2) = –3.2 + 1.2 = –2
a₃ − a₂ = –5.2 − (–3.2) = –5.2 + 3.2 = –2
a₄ − a₃ = –7.2 − (–5.2) = –7.2 + 5.2 = –2
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = –2
Three more terms
a₅ = a₄ + d = –7.2 + (–2) = –9.2
a₆ = a₅ + d = –9.2 + (–2) = –11.2
a₇ = a₆ + d = –11.2 + (–2) = –13.2
Answer: It is an AP series. Next three terms are –9.2, –11.2, –13.2
(iv) – 10, – 6, – 2, 2, . . .
Given sequence
–10, –6, –2, 2, …
Checking the differences
a₂ − a₁ = –6 − (–10) = –6 + 10 = 4
a₃ − a₂ = –2 − (–6) = –2 + 6 = 4
a₄ − a₃ = 2 − (–2) = 2 + 2 = 4
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = 4
Three more terms
a₅ = a₄ + d = 2 + 4 = 6
a₆ = a₅ + d = 6 + 4 = 10
a₇ = a₆ + d = 10 + 4 = 14
Answer: It is an AP series. Next three terms are 6, 10, 14
(v) 3, 3 + √2, 3 + 2√2, 3 + 3√2, …
Given sequence
3, 3 + √2, 3 + 2√2, 3 + 3√2, …
Checking the differences
a₂ − a₁ = (3 + √2) − 3 = √2
a₃ − a₂ = (3 + 2√2) − (3 + √2) = 2√2 − √2 = √2
a₄ − a₃ = (3 + 3√2) − (3 + 2√2) = 3√2 − 2√2 = √2
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = √2
Three more terms
a₅ = a₄ + d = (3 + 3√2) + √2 = 3 + 4√2
a₆ = a₅ + d = (3 + 4√2) + √2 = 3 + 5√2
a₇ = a₆ + d = (3 + 5√2) + √2 = 3 + 6√2
Answer: It is an AP series. Next three terms are 3 + 4√2, 3 + 5√2, 3 + 6√2
(vi) 0.2, 0.22, 0.222, 0.2222, . . .
Given sequence
0.2, 0.22, 0.222, 0.2222, …
Checking the differences
a₂ − a₁ = 0.22 − 0.2 = 0.02
a₃ − a₂ = 0.222 − 0.22 = 0.002
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: Not an AP
(vii) 0, – 4, – 8, –12, . . .
Given sequence
0, −4, −8, −12, …
Checking the differences
a₂ − a₁ = −4 − 0 = −4
a₃ − a₂ = −8 − (−4) = −8 + 4 = −4
a₄ − a₃ = −12 − (−8) = −12 + 8 = −4
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = −4
Three more terms
a₅ = a₄ + d = −12 + (−4) = −16
a₆ = a₅ + d = −16 + (−4) = −20
a₇ = a₆ + d = −20 + (−4) = −24
Answer: It is an AP series. Next three terms are −16, −20, −24
(viii) –1/2, –1/2, –1/2, –1/2, …
Given sequence
−1/2, −1/2, −1/2, −1/2, …
Checking the differences
a₂ − a₁ = −1/2 − (−1/2) = 0
a₃ − a₂ = −1/2 − (−1/2) = 0
a₄ − a₃ = −1/2 − (−1/2) = 0
Since the differences are the same (all 0), the sequence is an Arithmetic Progression (AP).
Common difference (d) = 0
Three more terms
a₅ = a₄ + d = −1/2 + 0 = −1/2
a₆ = a₅ + d = −1/2 + 0 = −1/2
a₇ = a₆ + d = −1/2 + 0 = −1/2
Answer: It is an AP series. Next three terms are −1/2, −1/2, −1/2
(ix) 1, 3, 9, 27, . . .
Given sequence
1, 3, 9, 27, …
Checking the differences
a₂ − a₁ = 3 − 1 = 2
a₃ − a₂ = 9 − 3 = 6
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: Not an AP
(x) a, 2a, 3a, 4a, . . .
Given sequence
a, 2a, 3a, 4a, …
Checking the differences
a₂ − a₁ = 2a − a = a
a₃ − a₂ = 3a − 2a = a
a₄ − a₃ = 4a − 3a = a
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = a
Three more terms
a₅ = a₄ + d = 4a + a = 5a
a₆ = a₅ + d = 5a + a = 6a
a₇ = a₆ + d = 6a + a = 7a
Answer: It is an AP series. Next three terms are 5a, 6a, 7a
(xi) a, a2, a3, a4, …
Given sequence
a, a², a³, a⁴, …
Checking the differences
a₂ − a₁ = a² − a = a(a − 1)
a₃ − a₂ = a³ − a² = a²(a − 1)
Since these differences depend on powers of a and are not constant (unless a = 1 or a = 0), the sequence is not an Arithmetic Progression (AP) in general. Try a = 2.
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: Not an AP
(xii) √2, √8, √18, √32, . . .
Given sequence
√2, √8, √18, √32, . . .
which is same as
√2, 2√2, 3√2, 4√2, …
Checking the differences
a₂ − a₁ = 2√2 − √2 = √2
a₃ − a₂ = 3√2 − 2√2 = √2
a₄ − a₃ = 4√2 − 3√2 = √2
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = √2
Three more terms
a₅ = a₄ + d = 4√2 + √2 = 5√2
a₆ = a₅ + d = 5√2 + √2 = 6√2
a₇ = a₆ + d = 6√2 + √2 = 7√2
Answer: It is an AP series. Next three terms are 5√2, 6√2, 7√2
(xiii) √3, √6, √9, √12, . . .
Given sequence
√3, √6, √9, √12, …
Checking the differences
a₂ − a₁ = √6 − √3 ≈ 2.449 − 1.732 = 0.717
a₃ − a₂ = √9 − √6 = 3 − 2.449 = 0.551
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: Not an AP
(xiv) 12, 32, 52, 72, . . .
Given sequence
Checking the differences
Since a₂ − a₁ ≠ a₃ − a₂, the sequence is not an Arithmetic Progression (AP).
Answer: Not an AP
(xv) 12, 52, 72, 73, . . .
Given sequence
1, 25, 49, 73, …
Checking the differences
a₂ − a₁ = 25 − 1 = 24
a₃ − a₂ = 49 − 25 = 24
a₄ − a₃ = 73 − 49 = 24
Since the differences are the same, the sequence is an Arithmetic Progression (AP).
Common difference (d) = 24
Three more terms
a₅ = a₄ + d = 73 + 24 = 97
a₆ = a₅ + d = 97 + 24 = 121
a₇ = a₆ + d = 121 + 24 = 145
Answer: It is an AP series. Next three terms are 97, 121, 145.