NCERT Exemplar Class 10 Maths Solutions Chapter 5 Arithmetic Progressions

NCERT Exemplar Class 10 Maths Solutions Chapter 5 Arithmetic Progressions

Exercise 5.1 Multiple Choice Questions (MCQs)

Question 1:
In an AP, if d = – 4, n = 7 and a_n = 4, then a is equal to
(a) 6                             (b) 7                        (c) 20                            (d) 28
Solution:

Question 2:
In an AP, if a = 3.5, d = 0 and n = 101, then a_n will be
(a) 0                             (b) 3.5                     (c) 103.5                       (d) 104.5
Solution:
(b) For an AP a_n = a + (n – 1)d= 3.5+ (101 – 1 )x 0                        [by given conditions]
∴                            = 3.5

Question 3:
The list of numbers – 10, – 6, – 2, 2,… is
(a) an AP with d = -16
(b) an AP with d = 4
(c) Fan AP with d = – 4
(d) not an AP
Solution:
(b) The given numbers are -10,-6,-2, 2…………………..
Here, a., = -10, a₂ = -6, a₃ = -2 and a₄ = 2………

Each successive term of given list has same difference i.e., 4.
So, the given list forms an AP with common difference, d=4.

Question 4:
The 11th term of an AP – 5,, 0,…
(a)-20                        (b)  20                        (c) -30                         (d) 30
Solution:
Given AP,- 5,, 0,

Question 5:
The first four terms of an AP whose first term is – 2 and the common difference is-2 are
(a) -2,0,2, 4
(b) -2, 4, -8,16
(c) -2,-4,-6,-8
(d) -2, – 4, -8, -16
Solution:
(c) Let the first four terms of an AP are a, a + d, a + 2d and a + 3d.
Given, that first term, a = – 2 and common difference, d = – 2, then we have an AP as follows
-2, – 2 – 2, – 2 + 2(-2), – 2 + 3(-2)
= – 2, – 4, – 6, – 8

Question 6:
The 21st term of an AP whose first two terms are – 3 and 4, is
(a) 17                (b) 137                       (c) 143                      (d)-143
Solution:
(b) Given, first two terms of an AP are a = – 3 and a + d = 4.
⇒                                          – 3 + d = 4

Question 7:
If the 2nd term of an AP is 13 and 5th term is 25, what is its 7th term?
(a) 30                 (b) 33                 (c) 37                   (d) 38
Solution:
(b) Given, a₂ = 13 and a₅ = 25

Question 8:
Which term of an AP : 21, 42, 63, 84,… is 210?
(a) 9th                         (b) 10th                    (c)11th                      (d) 12th
Solution:
(b) Let nth term of the given AP be 210

Hence, the 10th term of an AP is 210.

Question 9:
If the common difference of an AP is 5, then what is a₁₈ – a₁₃?
(a) 5                            (b) 20                       (c) 25                        (d) 30
Solution:
(c) Given, the common difference of AP i.e., d = 5

Question 10:
What is the common difference of an AP in which a₁₈ – a₁₄ = 32?
(a) 8                            (b) -8                        (c) – 4                        (d) 4
Solution:

Question 11:
Two APs have the same common difference. The first term of one of these is – 1 and that of the other is – 8. The difference between their 4th terms is
(a) -1                           (b) -8                        (c) 7                          (d) -9
Solution:
(c) Let the common difference of two APs are d₁ and d₂, respectively.
Bycondition,                          d₁ =d₂ =d                                                                      …(i)
Let the first term of first AP (a₁) = -1
and the first term of second AP (a₂) = – 8
We know that, the nth term of an AP, T₁ = a + (n – 1) d
∴4th term of first AP, T₄ = a, + (4 – 1)d = – 1 + 3d .
and 4th term of second AP, T’₄ = a₂ + (4 – 1)d = – 8 + 3d
Now, the difference between their 4th terms is i.e.,
|T4 -T’4|= (-1 + 3d)-(-8+3d)
= -1+ 3d + 8 – 3d = 7
Hence, the required difference is 7.

Question 12:
If 7 times the 7th term of an AP is equal to 11 times its 11th term, then its 18th term will be
(a) 7                           (b) 11                         (c) 18                     (d) 0
Solution:
(d) According to the question,

Question 13:
The 4th term from the end of an AP – 11, -8,-5,…, 49 is
(a) 37                 (b) 40                     (c)43                      (d) 58
Solution:
(b) We know that, the n th term of an AP from the end is

From Eq, (i), a₄ = 49 – (4 -1) 3 = 49 – 9 = 40

Question 14:
The famous mathematician associated with finding the sum of the first 100 natural
numbers is
(a)’Pythagoras
(b) Newton
(c) Gauss
(d) Euclid
Solution:
(c) Gauss is the famous mathematician associated with finding the sum of the first 100 natural numbers i,e.,      1,2, 3……………..100.

Question 15:
If the first term of an AP is – 5 and the common difference is 2, then the
sum of the first 6 terms is
(a) 0                      (b) 5                            (c) 6                           (d) 15
Solution:

Question 16:
The sum of first 16 terms of the AP 10, 6, 2, … is
(a)-320                      (b) 320                       (c)-352                      (d)-400
Solution:
(a) Given, AP is 10, 6, 2,…

Question 17:
In an AP, if a = 1, an = 20 and Sn = 399, then n is equal to
(a) 19                       (b) 21                         (c) 38                         (d) 42
Solution:

Question 18:
The sum of first five multiples of 3 is
(a) 45                         (b) 55                         (c) 65                        (d) 75
Solution:
(a) The first five multiples of 3 are 3, 6, 9,12 and 15.
Here, first term, a = 3, common difference, d = 6-3 = 3 and number of terms, n = 5

Exercise 5.2 Very Short Answer Type Questions

Question 1:
Which of the following form of an AP ? Justify your answer.

Solution:

Question 2:
Justify whether it is true to say that -1, , -2, … forms an AP as
a₂ – a₁ = a₃ -a₂

Solution:
False

Clearly, the difference of successive terms is not same, all though, a₂ – a₁ = a₃ -a₂
but a₃ – a₂ a₄ – a₃, therefore it does not form an AP.

Question 3:
For the AP -3, – 7, – 11,… can we find directly a3₀ – a₂₀ without actually finding a₃₀ and
a₂₀? Give reason for your answer.
Solution:
True

Question 4:
Two AP’s have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms? Why?
Solution:
Let the same common difference of two AP’s isd, Given that, the first term of first AP and second AP are 2 and 7 respectively, then the AP’s are
2,2 + d,2 + 2d,2 + 3d,.,.
and                                        7,7+ d, 7 +2d, 7+3d,…
Now, 10th terms of first and second AP’s are 2 + 9d and 7 + 9 d, respectively.
So, their difference is 7 + 9d – (2 + 9d) = 5
Also, 21st terms of first and second AP’s are 2 + 20d and 7 + 20d, respectively.
So, their difference is 7 + 20d – (2 + 9d) = 5
Also, if the a„ and b_n are the nth terms of first and second AP.
Then, b_n -a_n = [7 + (n-1)d)] – [2 + (n-1)d] = 5
Hence, the difference between any two corresponding terms of such AP’s is the same as the difference between their first terms.

Question 5:
Is 0 a term of the AP 31, 28, 25,…? Justify your answer.
Solution:
Let 0 be the nth term of given AP i.e., a_n = 0.
Given that, first term a = 31, common difference, d = 28 – 31 = – 3
The nth terms of an AP, is

Since, n should be positive integer. So, 0 is not a term of the given AP.

Question 6:
The taxi fare after each km, when the fare is ₹ 15 for the first km ahd ₹ 8 for each additional km, does not form an AP as the total fare (in ₹) after each km is 15, 8, 8, 8, … . Is the statement true? Give reasons.
Solution:
No, because the total fare (in ?) after each km is
15,(15+ 8), (15 + 2 x 8), (15+ 3 x 8),…= 15,23, 31, 39,…

Since, all the successive terms of the given list have same difference i.e., common difference = 8
Hence, the total fare after each km form an AP.

Question 7:
In which of the following situations, do the lists of numbers involved form an AP? Give reasons for your answers.
(i)  The fee charged from a student every month by a school for the whole session, when the monthly fee is ₹ 400.
(ii) The fee charged every month by a school from classes I to XII, When the monthly fee for class I is ₹ 250 and it increase by ₹ 50 for the next higher class.
(iii) The amount of money in the account of Varun at the end of every year when ₹ 1000 is deposited at simple interest of 10% per annum.
(iv) The number of bacteria in a certain food item after each second, when they double in every second.
Solution:
(i) The fee charged from a student every month by a school for the whole session is
400, 400, 400, 400,…
which form an AP, with common difference (d) = 400-400 = 0
(ii) The fee charged month by a school from I to XII is
250, (250 + 50), (250 + 2 x 50), (250 + 3 x 50),…
i.e.,                                     250,300,350,400,…
which form an AP, with common difference (d) = 300 – 250 = 50

Since, the difference between each successive term is not same, So, the list does form an AP

Question 8:
Justify whether it is true to say that the following are the nth terms of an
^AP.
(i) 2n – 3                     (ii) 3n² + 5                         (iii) 1 + n + n²

Solution:

Exercise 5.3 Short Answer Type Questions

Question 1:
Match the AP’s given in column A with suitable common differences given in column B.

Solution:

Question 2:
Verify that each of the following is an AP and then write its next three terms.

Solution:

Question 3:
Write the first three terms of the AP’s, when a and d are as given below
(i) a = ,d =                                          (ii) a = -5,d = -3                                       (iii) a = √2, d = 
Solution:

Question 4:
Find a, b and c such that the following numbers are in AP, a, 7, b, 23 and c.
solution :
Since a, 7, b, 23 and c are in AR

Question 5:

Determine the AP whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.
Solution:
Let the first term of an AP be a and common difference d.
Given, a₅ = 19and a₁₃ – a₈ = 20                                                                                     [given]

Question 6:
The 26th, 11th and the last terms of an AP are, 0,3 and , respectively.
Find the common difference and the number of terms.
Solution:
Let the first term, common difference and number of terms of an AP are a, d and n, respectively.
We know that, if last term of an AP is known, then

Question 7:
The sum of the 5th and the 7th terms of an AP is 52 and the 10th term is 46. Find the AP.
Solution:
Let the first term and common difference of AP are a and d, respectively.
According to the question,

Question 8:
Find the 20th term of the AP whose 7th term is 24 less than the 11th term, first term being 12.
Solution:
Let the first term, common difference and number of terms of an AP are a,d and n, respectively,             Given that, first term (a) = 12.
Now by condition,

Hence, the required 20th term of an AP is 126.

Question 9:
If the 9th term of an AP is zero, then prove that its 29th term is twice its 19th term.
Solution:
Let the first term, common difference and number of terms of an AP are a, d and n respectively.

Hence, its 29th term is twice its 19th term.

Question 10:
Find whether 55 is a term of the AP 7, 10, 13, … or not. If yes, find which term it is.
Solution:
Yes, let the first term, common difference and the number of terms of an AP are a, d and n respectively.
Let the nth term of an AP be 55. i.e., T_n = 55.

Hence, 17th term of an AP is 55.

Question 11:
Determine k, so that k² + 4k + 8, 2k² + 3k + 6 and 3k² + 4k + 4 are three consecutive terms of an AP.
Solution:
Since, k² + 4k + 8,2k² + 3k + 6 and 3k² + 4k + 4 are consecutive terms of an AP
∴                    2k² + 3k + 6 – (k² + 4k + 8) = 3k² + 4k + 4 – (2k² + 3k + 6) = Common difference

Question 12:
Split 207 into three parts such that these are in AP and the product of the two smaller parts is 4623.
Solution:
Let the three parts of the number 207 are (a – d), a and (a + d), which are in AP
Now, by given condition,

Hence, required three parts are 67, 69, 71.

Question 13:
The angles of a triangle are in AP. The greatest angle is twice the least. Find all the angles of the triangle.
Solution:
Given that, the angles of a triangle are in AR
Let A, B and C are angles of a Δ ABC.

Hence, the required angles of triangle are 80°, 60° and 40°.

Question 14:
If the nth terms of the two AP’s 9,7,5,……….. and 24,21,18,…. are the same, then find the value of n.Also,that term.
Solution:
Let the first term, common difference and number of terms of the AP 9, 7, 5,…are a₁,d₁ and n₁ respectively.
i.e., first term (a₁) = 9and common difference (d₁)= 7 – 9 = – 2.

Hence, the value of n is 16 and that term i.e., nth term is -21.

Question 15:
If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7th and 14th terms is -3, then find the 10th term.
Solution:
Let the first term and common difference of an AP are a and d, respectively.
According to the question,

Question 16:
Find the 12th term from the end of the AP – 2, – 4, – 6,……….,-100.
Solution:
Given AP -2, -4, -6,…,-100
Here, first term (a) = – 2, common difference (d) = – 4 – (-2) = -2 and the last term
(l) = -100.
We know,that, the nth term a„ of an AP from the end is a„ = l – (n – 1)d, where l is the last term and d is the common difference,
∴12th term from the end,
a12 =-100-(12-1)(-2)
= -100+ (11) (2) = – 100 + 22 = – 78.
Hence, the 12th term from the end is -78

Question 17:
Which term of the AP 53, 48, 43, … is the first negative term?
Solution:
Given AP is 53, 48, 43, …
Whose, first term (a) = 53 and common difference (d) = 48 – 53 = – 5
Let nth term of the AP be the first negative term.

Question 18:
How many numbers lie between 10 and 300, which divided by 4 leave a remainder 3?
Solution:
Here, the first number is 11, which divided by 4 leave remainder 3 between 10 and 300. Last term before 300 is 299, which divided by 4 leave remainder 3.
∴                                                         11,15,19,23…. 299
Here, first term (a) = 11, common difference d = 15 -11 = 4

Question 19:
Find the sum of the two middle most terms of an AP
,-1,,………..,4
Solution:

Question 20:
The first term of an AP is – 5 and the last term is 45. If the sum of the terms of the AP is 120, then find the number of terms and the common difference.
Solution:
Let the first term, common difference and the number of terms of an AP are a, d and n respectively.
Given that, first term (a) = – 5 and last term (l) = 45
Sum of the terms of the AP = 120 ⇒ S_n = 120
We know that, if last term of an AP is known, then sum of n terms of an AP is,

So, the common difference is 10.
Hence, number of terms and the common difference of an AP are 6 and 10 respectively.

Question 21:
Find the sum

Solution:

Question 22:
Which term of the AP – 2, – 7, – 12,… will be – 77 ? Find the sum of this AP upto the term – 77.
Solution:
Given, AP -2,-7,-12,…
Let the nth term of an AP is – 77.
Then, first term (a) = – 2 and common difference (d) = – 7 – (- 2) = – 7 + 2 = – 5.
∴nth term of an AP, T_n = a + (n – 1)d

Hence, the sum of this AP upto the term – 77 is – 632.

Question 23:
If a_n = 3 — 4n, then show that a_1,a₂, a₃, …form an AP. Also, findS₂₀.
Solution:
Given that, nth term of the series is an = 3 – 4n                                                                            …(i)

the required sum of 20 terms i.e., S₂₀ is – 780.

Question 24:
In an AP, if s_n =n (4n + 1), then find the AP.
Solution:
We know that, the n th term of an AP is


Hence, the required AP is 5,13, 21,…

Question 25:
In an AP, if s_n = 3n² + 5n and a_k = 164, then find the value of k.
Solution:

Question 26:
If s_n denotes the sum of first n terms of an AP, then prove that   s₁₂ =3(s₈-s₄)
Solution:

Question 27:
Find the sum of first 17 terms of an AP whose 4th and 9th terms are – 15 and – 30, respectively.
Solution:
Let the first term, common difference and the number of terms in an AP are a, d and n,respectively


Hence,the required sum of first 17 terms  of an AP is -510

Question 28:
If sum of first 6 terms of an AP is 36 and that of the first 16 terms is 256, then find the sum of first 10 terms.
Solution:
Let a and d be the first term and common difference, respectively of an AP

Hence, the required sum of first 10 terms is 100.

Question 29:
Find the sum of all the 11 terms of an AP whose middle most term is 30.
Solution:
Since, the total number of terms (n)=11                                                                               [odd]

Question 30:
Find the sum of last ten terms of the AP 8, 10, 12,…, 126.
solution :
For finding, the sum of last ten terms, we write the given AP in reverse order.
i.e., 126,124,122… 12,10,8
Here, first term (a) = 126, common difference, (d) = 124-126=-2

Question 31:
Find the sum of first seven numbers which are multiples of 2 as well as of 9.
Solution:
For finding, the sum of first seven numbers which are multiples of 2 as well as of 9. Take LCM of 2 and 9 which is 18.
So, the series becomes 18, 36, 54,…
Here, first term (a) = 18, common difference (of) = 36 -18 = 18

Question 32:
How many terms of the AP – 15, – 13, -11, … are needed to make the sum 55?
Solution:
Let n number of terms are needed to make the sum – 55.
Here, first term (a) = -15, common difference (d) = -13+15 = 2

Hence, either 5 and 11 terms are needed to make the sum – 55.

Question 33:
The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is – 30 and the. common difference is 8. Find n.
Solution:
Given that, first term of the first AP (a) = 8
and common difference of the first AP (d) = 20
Let the number of terms in first AP be n.


Hence, the required value of n is 11.

Question 34:
Kanika was given her pocket money on Jan 1st, 2008. She puts ₹ 1 on day 1, ₹ 2 on day 2, ₹ 3 on day 3 and continued doing so till the end of the month, from this money into her piggy bank she also spent ₹ 204 of her pocket money, and found that at the end of the month she still had ₹ 100 with her. How much was her pocket money for the month?
Solution:
Let her pocket money be  ₹ x.
Now, she takes  ₹ 1 on day 1,  ₹ 2 on day 2,  ₹ 3 on day 3 and so on till the end of the month, from this money.
i.e.,                                          1 + 2 + 3+ 4+ … + 31.
which form an AP in which terms are 31 and first term (a) = 1, common difference (d) = 2 — 1 = 1 .
Sum of first 31 terms = S₃₁

Hence, ₹ 800 was her poket money for the month

Question 35:
Yasmeen saves  ₹ 32 during the first month,  ₹ 36 in the second month and ₹ 40 in the third month. If she continues to save in this manner, in how many months will she save  ₹ 2000 ?
Solution:
Given that,
Yasmeen, during the first month, saves = ₹ 32
During the second month, saves = ₹ 36
During the third month, saves = ₹ 40
Let Yasmeen saves ₹ 2000 during the n months.
Here, we have arithmetic progression 32, 36, 40,…
First term (a) = 32, common difference (d) = 36 – 32 = 4
and she saves total money, i.e., S_n = ₹ 2000
We know that, sum of first n terms of an AP is,

Hence, in 25 months will she save ₹ 2000.                                                        [Since, month cannot be negative]

Exercise 5.4 Long Answer Type Questions

Question 1:
The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.
Solution:
Let the first term, common difference and the number of terms of an AP are a, d and n, respectively.


Hence, the required sum of its first twenty terms is 970.

Question 2:
Find the
(i)   sum of those integers between 1 and 500 which are multiples of 2 as well as of 5.
(ii)  sum of those integers from 1 to 500 which are multiples of 2 as well as of 5.
(iii) sum of those integers from 1 to 500 which are multiples of 2 or 5.
Solution:
(i) Since, multiples of 2 as well as of 5 = LCM of (2, 5) = 10
∴  Multiples of 2 as well as of 5 between 1 and 500 is 10, 20, 30. 490
which form an AP with first term (a) = 10 and common difference (d) = 20 -10 = 10
nth term a_n =Last term (/) = 490
∴ Sum of n terms between 1 and 500,

Question 3:
The eighth term of an AP is half its second term and the eleventh term exceeds one-third of its fourth term by 1. Find the 15th term.
Solution:
Let a and d be the first term and common difference of an AP respectively.

Question 4:
An AP consists of 37 terms. The sum of the three middle most terms and the sum of the last three terms is 429. Find the AP.
Solution:
Since, total number of terms                                                                                                                 [Odd]

Question 5:
Find the sum of the integers between 100 and that are
(i) divisible by 9.                                                (ii) not divisible by 9.
Solution:
(i) The numbers (integers) between 100 and 200 which is divisible by 9 are 108, 117, 126, … 198.
Let n be the number of terms between 100 and 200 which is divisible by 9.

Hence, required sum of the integers between 100 and 200 that are divisible by 9 is 1683.
(ii) The sum of the integers between 100 and 200 which is not divisible by 9 = (sum of total numbers between 100 and 200) – (sum of total numbers between 100 and 200 which is divisible by 9).

Hence, the required sum is 13167.

Question 6:
The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term and also the ratio of the sum of the first five terms to the sum of the first 21 terms.
Solution:
Let a arid d be the first term and common difference of an AP

So, ratio of the sum of the first five terms to the sum of the first 21 terms
S5 :S21 = 30 d :294 d = 5:49

Question 7:
Show that the sum of an AP whose first term is a, the second term b and the last term c, is equal to  
Solution:
Given that, the AP is a, b………..c
Here, first term = a, common difference = b – a

Question 8:
Solve the equation – 4 + (-1) + 2 +… + x = 437.
Solution:
Given equation is, – 4 -1 + 2 + … + x = 437                                                                                  …(i)
Here, – 4-1 + 2 + …+ x forms an AP with first term =-4, common difference =3,
a_n = l = x

Here, x cannot be negative, i.e., x ≠- 53
also, for x = -53, n will be negatuve which is not possible
Hence, the required value of x is 50.

Question 9:
Jaspal Singh repays his total loan of ₹ 118000 by paying every month starting with the first installment of ₹ 1000. If he increases the installment by ₹ 100 every month, what amount will be paid by him in the 30th installment? What amount of loan does he still have to pay after the 30th installment?
Solution:
Given that,
Jaspal singh takes total loan = ₹ 118000 He repays his total loan by paying every month.
His first installment = ₹ 1000
Second installment = 1000 + 100 = ₹ 1100
Third installment = 1100 + 100 = ₹ 1200 and so on

Question 10:
The students of a school decided to beautify the school on the annual day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags.
Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance she did cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?
Solution:
Given that, the students of a school decided to beautify the school oh the annua] day by fixing colourful flags on the straight passage of the school.
Given that, the number of flags = 27 and distance between each flag = 2 m.
Also, the flags are stored at the position of the middle most flag i.e., 14th flag and Ruchi wds given the responsibility of placing the flags. Ruchi kept her books, where the flags were stored i.e., 14th flag and she could carry only one flag at a time.
Let she placed 13 flags into her left position from middle most flag i.e., 14th flag. For placing second flag and return his initial position distance travelled = 2+ 2= 4 m.
Similarly, for placing third flag and return his initial position, distance travelled =t 4+ 4= 8 m For placing fourth flag and return his initial position, distance travelled = 6+ 6= 12 m.
For placing fourteenth flag and return his initial position, distance travelled ,
= 26+ 26= 52 m
Proceed same manner into her right position from middle most flag
i.e., 14th flag. Total distance travelled in that case = 52 m
Also, when Ruchi placed the last flag she return his middle most position and collect her books. This distance also included in placed the last flag.
So, these distances form a series.
4+ 8+ 12 + 16+ …+ 52                                                    [for left]
and                                          4+8+12 +16+… + 52                                                  [for right]

Hence, the required is 728 m in which she did cover in completing this job and returning back to collect her books.
Now, the maximum distance she travelled carrying a flag = Distance travelled by Ruchi
during placing the 14th flag in her left position or 27th flag in her right position
= (2 + 2 + 2 + … + 13 times)
= 2 x 13 = 26m
Hence, the required maximum distance she travelled carrying a flag is 26 m.

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