CBSE Class 10 Mathematics Arithmetic Progressions VBQs

Question. If an AP has a1 = 1, an = 20 and Sn = 399, then the value of n is
(a) 20
(b) 32
(c) 38
(d) 40
Answer : C

Question. A man receives Rs. 60 for the first week and Rs. 3 more each week than the preceding week. How much does he earn by the 20th week?
(a) Rs. 1760
(b) Rs. 1770
(c) Rs. 1780
(d) Rs. 1790
Answer : B

Question. Two APs have the same common difference. The first term of one of these is –1 and that of the other is – 8. Then the difference between their 4th terms is
(a) –1
(b) – 8
(c) 7
(d) –9
Answer : C

Question. If an = 5n – 4 is a sequence, then a₁₂ is
(a) 48
(b) 52
(c) 56
(d) 62
Answer : C

Question. If a_n = 3n – 2, then the value of a₇ + a₈ is
(a) 39
(b) 41
(c) 47
(d) 53
Answer : B

Question. The second term of the sequence defined by a_n = 3_n + 2 is
(a) 2
(b) 4
(c) 6
(d) 8
Answer : D

Assertion-Reason Type Questions

In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion (A): The arrangement of numbers, i.e., – 4, 16, – 64, 256, – 1024, 4096, ... form a sequence.
Reason (R): An arrangement of numbers which are arranged in a definite order according to some rule, is called a sequence.
Answer : A

Question. Assertion (A): Sequence 1, 5, 9, 13, 17, 21, ... is a finite sequence.
Reason (R): A sequence with finite number of terms or numbers is called a finite sequence.
Answer : D

Answer the following:

Question. Write down the first six terms of each of the following sequences, whose general terms are:
(a) a_n = 5n – 3
(b) a_n = (– 1)ⁿ ⋅ 2²ⁿ
(c) a_n = 2n+1/n+2
(d) a_n = (–1)^{n – 1} ⋅ n²
Answer : (a) 2, 7, 12, 17, 22, 27
(b) –4, 16, –64, 256, –1024, 4096
(c) 1, 5/4, 7/5, 3/2, 11/7, 13/8
(d) 1, –4, 9, –16, 25, –36

Question. Find the 10^th term of the sequence defined by a_n = (–1)2^{n – 1} ⋅ 5_n.
Answer : –9765625

Question. Find the difference between the 12^th term and 10th term of the sequence whose general term is given by a_n = 5n – 1.
Answer : 10
 

Value Based Question

Question. Write first four terms of the AP when first term is -2 and common difference is 0
Answer : 2.-2,-2,-2

Question. Subba Rao stared work in 1995 at an annual salary of Rs 5000 and received an increament of Rs 200 each year . in Which year did his income reach Rs. 7000.
Answer : 11th years

Question. Which terms of an AP : 3,8,13,18,….is 78
Answer : 16th term

Question. Which term of the AP: 23, 21, 19,.....is first negative term? Also, find this term.
Answer : 13th , -1

Question. How many two digit numbers are divisible by 3?
Answer : 30

Question. Find the number of all two-digit natural numbers which are divisible by 6.
Answer : 15

Question.The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 each additional km find fare after 4 km.
Answer : 39

Question. A man saves Rs. 10 on the first day of the month Rs 20 on the second day Rs. 30 on the third day and so on What will be saving in 30 days
Answer : Rs 4500

Question. 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 ?
Answer : n = 10

Question. How many natural numbers are there between 200 and 500, which are divisible by 7?
Answer : 43

Question. The sum of the first seven terms of an AP is 182. If its 4th and the 17^th terms are in the ratio 1: 5, find the AP.
Answer : 2,10,18,...

Question. A thief after committing a theft runs a uniform speed of 50m/minute. After 2 minute, a policeman runs to catch him. He goes 60m in first minute and increases his speed by 5m/minute every succeeding minute. After how many minutes, the police will catch the thief.
Answer : 7 minutes

Question. If its 7th term is 4, find the AP.
Answer :- 16, 14, 12, 10……

Question. If k, 2k−1 and 2k+1 are three consecutive terms of an A. P. , then find the value of k.
Answer : 3

Question. Three numbers are in AP. If the sum of these numbers be 27 and the product 648, find the numbers.
Answer : 6, 9, 12 or 12,9,6

Question. The sum of first m terms of an AP is 4m²− m. If its n^th term is 107, find the value of n. Also find the 21st term of this of this AP.
Answer : n =14 , 163

Question. Ragav buys a shop for Rs. 120000. He pays the amount in case and agree to pay the balance in 12 annual installment of Rs. 5000 each if the rate of interest is 12 % and he pays with the installment the interest due on the unpaid amount . find the total cost of shop.
Answer : Rs. 166800.

Question. In an AP, the sum of its first ten terms is −80 and the sum its next ten terms is −280.Find the AP.
Answer : 1,−1,−3,−5,…

Question. Determine the AP whose5th term is 19 and the difference of the 8th term from the 13th term is 20.
Answer : 3,7,11,15

Question. Find the number of three-digit natural numbers which are divisible by 11.
Answer : 81 13. The 5th term of an AP exceeds its 12th term by

Question. If an = 9 – 5ⁿ find sum of 15th terms of an AP
Answer : - 66

Question. A man is in the habit of drinking daily. Amounts spent on drinking are as follows:
Rs. 25, Rs. 40, Rs 55, Rs 70 . . . in successive days.What will be total amount spent by him in 30 successive days? Write any two disadvantages of drinking.
Answer : Rs.7275

Short Ansser type Questions

Question. How many terms of the AP 27, 24, 21, … should be taken so that their sum is zero.
Answer : The first term (a) = 27, The sum of first n terms (Sn) = 0 Common difference of the A.P. (d) = a2 – a1 = 24 – 27 = -3. On substituting the values in Sn, we get 0 = 𝑛/2[2(27) + (n − 1)( − 3)] , 0 = (n)[54 + (n – 1)(-3)] , 0 = (n)[54 – 3n + 3] ⟹ 0 = n [57 – 3n] Further we have, n = 0 Or, 57 – 3n = 0 ⟹ 3n = 57 ⟹ n = 57/3 = 19. The number of terms cannot be zero. Hence n=19

Question. Check whether -150 is a term of the AP: 11,8, 5, 2, …
Answer : 11, 8, 5, 2, …-150, a = 11, d = 8- 11 = -3, an = -150, 11 + (n – 1) (-3) = -150, 11 – 3n + 3 = -150 , -3n + 14 = -150 , -3n = -150 – 14, -3n = -164, 3n = 164 ∴ n = 164/3,
Here value of ‘n’ is not a positive integer. Hence -150 is not a term of the given AP

Question. The 4^th term of an AP is zero. Prove that 25^th term is three times its 11^th term.
Answer : a + 3d = 0 or a = -3d…(1)
,a₂₅ = a + 24d = -3d + 24d = 21d …(2)
a₁₁ = a + 10d = -3d + 10d = 7d ….(3)
From (2) and (3), we have 21d= 3x7d , a₂₅ = 3 x a₁₁ Hence proved.

Question. How many two-digit numbers are divisible by 3?
Answer : The Required A.P = 12, 15 ,18 .......... 99 ,First term (a) = 12, Last term = 99, Common difference = 3, nth term = a+(n-1)d,. Putting the values in the formula:=> 99 = 12+(n-1)3, 99-12 = 3(n-1), 87 = 3(n-1), 87/3= n-1, 29 = n-1, 29+1 = n, n = 30

Question. Which term of the AP. 20, 17, 14, .........; is the first negative term?
Answer : a_n <0 , 20 + (n-1)-3<0 ,20-3n+3<0,23-3n<0, 23<3n ,23/3<n, 7.6<n. Next natural number greater than 7.6 is 8.Hence 8th term is the first negative number.

Question. Which term of the AP 3,15, 27, 39, … is 132 more than its 54^th term?
Answer : a1 = 3, _a2= 15, d = 15 - 3 = 12, 54th term of the AP is a₅₄ = a + (54 - 1)d = 3 + 53 ×12 = 639, Let nth term of AP be 132 more than 54^th term ,We get, 132 + 639 = 771, aₙ = 771, 771 = 3 + (n - 1)12, 768 = (n - 1)12, (n - 1) = 64, n = 65, Therefore, the 65^th term will be 132 more than the 54^th term

Question. The sum of first n terms of an AP is given by S_n = 2n²+n. Then find its nth term .
Answer : The sum of the first n terms of an A.P. is given by Sn= 2n² +n ,
At n=1 , S₁= 2x1²+1= 3 , At n=2 , S₂= 2x2²+2=10 ,
Since a₁=S₁ ,S₂=a1+a₂,
So , a₁=3 ,a1+a2=10,⇒3+a₂=10 so a₂=7 ,d= 7-3=4,an= 3+(n-1)4=4n-1

Question. Find the sum of all odd numbers between 10 and 200.
Answer : Odd numbers between 10 and 200 are 11,13,15....199. a1 = 11, Last term l = 199, d = 2, an = a + (n-1) d, 199 = 11 + (n - 1) 2,199 - 11 = (n - 1) 2,188 = (n - 1) 2, 94 = n – 1,95 = n
Sum of n terms = 𝑛/2(a + l), =95/2 (11 + 199)= 9975

Question. If the sum of first n terms of an AP is n² find the 5^th term.
Answer : Given Sn = n²,we know , a_n = S_n - S(n -1), a₅ = S₅ - S(5 -1)= S₅ – S₄= 5₂-4²= 2⁵-16=9

Question. Find the middle term of the AP -11, -7, -3, …, 45.
Answer : Given AP is -11,-7,-3,......,45
Here a = -11, d = -7- -11=-7+11=4 and last term l = 45
45 = (-11) + (n - 1)4, 56 = (n - 1)4 , n - 1 = 14,
Therefore, n = 15 That is there are 15 terms.
Hence 8th term is the middle most term of the given AP a8 = a + 7d = (-11) + 7(4) = 17.
Thus the middle term is 17

Question. If an AP has 8 as the first term, -5 as the common difference and its first 3 terms are 8, A, B, then find A+B.
Answer : The first term of the AP =8,Common difference d = -5
Given that A is the second term, So, A = 8+(-5) = 8-5 = 3
Given that B is the third term So, B= 3+(-5) = 3-5 = -2
So (A+B) = 3+(-2) = 3-2 = 1

CASE STUDY QUESTIONS

Q1. 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. Write first four terms are in AP for the given situation.
Answer : 51,49, 47, 45…..

Question. What is the minimum number of days he needs to practice till his goal is achieved.
Answer : 11

Question. How many second takes after 5^th days .
Answer : 43 seconds

Question. A school decided to disburse Rs.4,000 as award to their students for punctuality. The most punctual student was to be awarded Rs.1200 and the next prizes were to be Rs. 200 less than the previous one.

CONTEXTUAL QUESTION 

(I) Find the number of students who received the award

VALUE BASED QUESTION 

(II) What value (any two) is highlighted in this context?

Answer: (II) KEY POINTS

* Punctuality

* Value for time

* Responsibility

* Commitment

* Self discipline

Question. In her piggy bank, Lakshmi dropped Re.1.00 on May1, Rs.1.75 on May 2, Rs. 2.50 on May 3 and so on until the last day of May. She decided to get a gift for the nearby old age home from her total deposit.

CONTEXTUAL QUESTION 

(I) How much did she drop in her piggy bank on May 19?

(II) What was her total deposit in her piggy bank for the month of May?

VALUE BASED QUESTION 

(III)Which value (any two) does she posses in this context?

Answer: (III) KEY POINTS

* Self discipline

* Charity

* Concern for elders

* Sharing and caring

 * Responsible citizenship

 *  Concern for future citizens

 * Caring

 *  Responsible

 *  Compassion

Question.There are 20 rows of seats on a concert hall: 25 seats are in the first row, 27 seats on the second row, 29 seats on the third row, and so on. Ms. Uma a student of XYZ school gave a music concert during the December holiday and collected Rs. 1,00,000 as a fund raiser for Flood victims, given the price per ticket is Rs. 2,300.

CONTEXTUAL QUESTION