CBSE Class 10 Arithmetic Progressions Sure Shot Questions Set 17

Your elder brother wants to buy a car and plans to take loan from a bank for purchasing the car. He repays his total loan of ₹ 1,18,000 by paying every month starting with the first installment of ₹ 1000. If he increases the instalment by ₹ 100 every month, answer the following :

Question. What is the first term of an A.P.?
(a) 1000
(b) 100
(c) 1800
(d) 118000
Answer: (a) 1000
Explanation :
As first installment is ₹ 1000. So, first term of A.P. is 1,000.

Question. What is the common difference of an A.P.?
(a) 1000
(b) 100
(c) 1800
(d) 118000
Answer: (b) 100
Explanation :
As every consecutive number differ by 100.

Question. The amount paid by him in \( 30^{th} \) installment is
(a) 3900
(b) 3500
(c) 3700
(d) 3600
Answer: (a) 3900
Explanation :
\( a = 1000, d = 100, n = 30 \)
Now, \( a_{30} = a + 29d \)
\( = 1000 + 29 \times 100 \)
\( = 3900 \)

Question. If total installments are 40 then what is the amount paid in the last installment?
(a) 4900
(b) 3900
(c) 5900
(d) 9400
Answer: (a) 4900
Explanation :
\( a = 1000, d = 100 \)
\( \therefore a_{40} = a + 39d \)
\( = 1000 + 39 \times 100 \)
\( = 4900 \)

Question. The ratio of the \( 1^{st} \) installment to the last installment is :
(a) 1 : 49
(b) 10 : 49
(c) 10 : 39
(d) 39 : 10
Answer: (b) 10 : 49
Explanation :
\( a = 1000 \)
\( a_{40} = 4900 \)
\( \frac{a}{a_{40}} = \frac{1000}{4900} = 10 : 49 \)

Aadita is celebrating her birthday. She invited her friends. She bought a packet of toffees/candies. She arranged the candies such that in the first row there are 3 candies, in second there are 5 candies, in third there are 7 candies and so on.

Question. Find the first term and common difference of A.P.
(a) \( a = 2, d = 3 \)
(b) \( a = 3, d = 2 \)
(c) \( a = 2, d = - 3 \)
(d) \( a = 3, d = - 2 \)
Answer: (b) \( a = 3, d = 2 \)
Explanation :
As A.P. = 3, 5, 7 ...
\( \therefore a = 3, d = 2 \)

Question. How many candies are placed in the \( 9^{th} \) row?
(a) 22
(b) 21
(c) 24
(d) 18
Answer: (b) 21
Explanation :
Since, \( a_n = a + (n - 1)d \)
\( a_n = 3 + 9 \times 2 = 21 \)

Question. Find the difference in number of candies placed in \( 7^{th} \) and \( 3^{rd} \) row.
(a) 8
(b) 10
(c) 12
(d) 14
Answer: (a) 8
Explanation :
\( a_7 - a_3 = a + 6d - a - 2d = 4d \)
\( = 4 \times 2 = 8 \)

Question. Find the number of candies in \( 12^{th} \) row.
(a) 21
(b) 30
(c) 25
(d) 19
Answer: (c) 25
Explanation :
Number of candies in \( 12^{th} \) row,
\( a_{12} = a + 11d \)
\( = 3 + 11 \times 2 = 25 \)

Question. Find the number of candies in \( 15^{th} \) row.
(a) 30
(b) 32
(c) 28
(d) 31
Answer: (d) 31
Explanation :
Number of candies in \( 15^{th} \) row,
\( a_{15} = a + 14d \)
\( = 3 + 14 \times 2 = 31 \)

Jack is much worried about his upcoming assessment on Arithmetic Progression. He was vigorously practicing for the exams but unable to solve some questions. One of these question is given below : If the \( 3^{rd} \) and \( 9^{th} \) term of an A.P. are 4 and – 8 respectively, then help jack in solving problem:

Question. What is the common difference of the AP?
(a) 2
(b) – 1
(c) – 2
(d) 4
Answer: (c) – 2
Explanation :
\( a_3 = a + 2d = 4 \)...(i)
\( a_9 = a + 8d = - 8 \)...(ii)
Solving equations (i) and (ii)
\( d = - 2 \)

Question. What is the first term of the A.P.?
(a) 6
(b) 2
(c) – 2
(d) 8
Answer: (d) 8
Explanation :
\( a_3 = a + 2d = 4 \)...(i)
\( a_9 = a + 8d = - 8 \)...(ii)
Solving equations (i) and (ii), we get
\( a = 8 \)

Question. Which term of the A.P. is – 160?
(a) \( 80^{th} \)
(b) \( 85^{th} \)
(c) \( 81^{st} \)
(d) \( 84^{th} \)
Answer: (b) \( 85^{th} \)
Explanation :
Let \( t_n = - 160 = a + (n - 1)d \)
\( \Rightarrow - 160 = 8 + (n - 1)(- 2) \)
\( \Rightarrow n = 85^{th} \text{ term} \)

Question. Which of the following is not the term of an A.P.
(a) – 123
(b) – 100
(c) 0
(d) – 200
Answer: (a) – 123
Explanation :
As, \( a_n = a + (n - 1)d \)
\( - 123 = 8 + (n - 1) \times (- 2) \)
\( \Rightarrow (n - 1) \times - 2 = - 131 \)
\( \Rightarrow n = \frac{-131}{-2} + 1 \)
Which is not an whole number.

Question. Which is the \( 75^{th} \) term of an A.P.?
(a) – 140
(b) – 102
(c) – 150
(d) – 158
Answer: (a) – 140
Explanation :
\( T_{75} = a + 74d \)
\( \therefore 8 + 74 \times (- 2) = - 140 \)

Self-Assessment

Question. The \( n^{th} \) term of an A.P. is \( 6n + 2 \). Find the common difference. 
Answer: 6.

Question. A sum of ₹ 1000 is invested at 8% S.I. per annum. Calculate the rate of interest at the end of 1, 2, 3, … years. Is the sequence of the interests an A.P. ? Find the interest at the end of 30 years. 
Answer: The interest after 1 year = ₹ 80
The interest after 2 years = ₹ 160
The interest after 3 years = ₹ 240
Yes, the sequence of the interests is an A.P.
The interest after 30 years = ₹ 2400.

Question. In a flower bed, there are 23 rose plants in the first row, 21 in the second row, 19 in the third row and so on. There are five plants in the last row. How many rows are there in the flower bed ? 
Answer: 10.

Question. A person started working in 1995 at a salary of ₹ 5000 per month with a yearly increment of ₹ 200. In which year did his salary reach ₹ 7000 per month ? 
Answer: In the year 2005.

Question. A housewife saved ₹ 5 in the first week of the year and thereafter increased her weekly savings by ₹ 1.75. After how many weeks will her weekly savings be ₹ 20.75 ? 
Answer: 10 weeks.

Question. Find the \( 12^{th} \) term from the end of the following arithmetic progression : 3, 8, 13, …, 253. 
Answer: 198.

Question. The sum of the \( 4^{th} \) and \( 8^{th} \) terms of an A.P. is 24 and the sum of the \( 6^{th} \) and \( 10^{th} \) term is 34. Find the first term and the common difference of the A.P. [NCERT]
Answer: – 0.5, 2.5.

Question. Find the A.P. whose third term is 16 and the seventh term exceeds its fifth term by 12. 
Answer: 4, 10, 16, 22, …

Question. Two A.P.s have the same common difference. The difference between their \( 100^{th} \) terms is 100. What is the difference between their \( 1000^{th} \) terms ? 
Answer: 100.

Question. A manufacturer of TV sets produced 600 units in the \( 3^{rd} \) year and 700 units in the \( 7^{th} \) year. Assuming that the production increases uniformly by a fixed number every year, find the production in (i) the first year, (ii) the \( 10^{th} \) year. 
Answer: (i) 550, (ii) 775.

Question. The contract of a construction job specifies a penalty for delay of completion beyond a certain date as follows : ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day and so on. How much does a delay of 30 days cost the contractor ? 
Answer: ₹ 27,750.

Question. A sum of ₹ 280 is to be used to award four prizes. If each prize after the first is ₹ 20 less than its preceding prize, find the value of each of the prizes. 
Answer: The first prize is ₹100, second prize is ₹ 80, third prize is ₹ 60 and fourth prize is ₹ 40.

Question. A football gallery is 50 m wide and each step is 0.5 m in length. If the height of each step increases by 0.25 m from the first, find the volume of concrete required to make a gallery of 15 steps. 
Answer: 11250 \( \text{m}^3 \).

Question. How many terms of the A.P. 9, 17, 25, … must be taken so that their sum is 636 ? 
Answer: 12.

Question. Find the sum of the first 15 terms of the series where \( t_n = 9 - 5n \). 
Answer: – 465.

Question. Find the sum of the first 51 terms of an A.P. whose \( 2^{nd} \) and \( 3^{rd} \) terms are 14 and 18 respectively. 
Answer: 5610.

Question. The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. 
Answer: 16, \( \frac{8}{3} \).

Question. Find the \( n^{th} \) term and the last term of an A.P. whose first term is 2, the common difference is 8 and the sum of all the terms is 90. 
Answer: \( 8n - 6, 34 \).

Question. If the sum of the first \( n \) terms of an A.P. is \( 4n - n^2 \), find the first term, second term and sum of the first two terms.
Answer: 3, 1, 4.

Question. The first term of an A.P. is 17 and the last term is 350. If the common difference is 9, how many terms are there and what is their sum ? 
Answer: 38, 6973.

Question. The sum of \( n \) terms of an A.P. is \( 5n - n^2 \). Find \( n^{th} \) term of this A.P.
Answer: \( 6 - 2n \).

Question. The sum of first \( n \) terms of an A.P. is \( 3n^2 + 4n \). Find the \( 25^{th} \) term of this A.P.
Answer: 157.