CBSE Class 10 Mathematics Arithmetic Progressions Worksheet Set 10

Key Notes

• Sequence: A set of numbers arranged in some definite order and formed according to some rules is called a sequence.
• Progression: The sequence that follows a certain pattern is called progression.
• Arithmetic Progression: A sequence in which the difference obtained by subtracting any term from its preceding term is constant throughout, is called an arithmetic sequence or arithmetic progression (A.P.). The general form of an A.P. is \( a, a + d, a + 2d, ... \) (a: first term common difference).
• General Term: If ‘a’ is the first term and ‘d’ is common difference in an A.P., then \( n^{th} \) term (general term) is given by \( a_n = a + (n - 1)d \).
• The formula \( a_n = a + (n - 1)d \) contains four quantities \( a_n \), \( a \), \( n \) and \( d \). Three quantities being given, the fourth can be find out by using the above relation.
• If only two quantities are given, two conditions (equations) in the problem should be given. Therefore, to determine these two unknowns, we have to solve both the conditions (equations) linearly.
• Sum of n Terms of an A.P.: If ‘a’ is the first term and ‘d’ is the common difference of an A.P., then sum of first n terms is given by \( S_n = \frac{n}{2} \{2a + (n - 1)d\} \). If ‘l’ is the last term of a finite A.P. then the sum is given by \( S_n = \frac{n}{2} \{a + l\} \).
• (i) If \( a_n \) is given, then common difference \( d = a_n - a_{n-1} \).
  (ii) If \( S_n \) is given, then \( n^{th} \) term is given by \( a_n = S_n - S_{n-1} \).
  (iii) If \( a, b, c \) is in A.P., then \( 2b = a + c \).
  (iv) If a sequence has \( n \) terms, its \( r^{th} \) term from the end \( = (n - r + 1)^{th} \) term from the beginning.

Question. How many two digit number are there which are divisible by 7 ?
(a) 13
(b) 14
(c) 15
(d) None of the options
Answer: (a) 13

Question. How many numbers are there between 103 and 750 which are divisible by 6 ?
(a) 125
(b) 108
(c) 107
(d) 113
Answer: (b) 108

Question. The sum of first 60 natural numbers is –
(a) 1830
(b) 1640
(c) 3660
(d) 1770
Answer: (a) 1830

Question. The sum of all 2 digit numbers is –
(a) 4750
(b) 4905
(c) 3776
(d) 4680
Answer: (b) 4905

Question. 23rd term of the A.P. 7, 5, 3, 1, ....... is –
(a) 51
(b) 37
(c) –37
(d) –51
Answer: (c) –37

Question. If (k + 1), 3k and (4k + 2) be any three consecutive terms of an A.P., then the value of k is –
(a) 3
(b) 0
(c) 1
(d) 2
Answer: (a) 3

Question. Which term of the A.P. 5, 8, 11, 14 .... is 320 ?
(a) 104th
(b) 105th
(c) 106th
(d) 64th
Answer: (c) 106th

Question. The 5th and 13th terms of an A.P. are 5 and – 3 respectively. The first term of the A.P. is –
(a) 1
(b) 9
(c) –15
(d) 2
Answer: (b) 9

Question. Which term of the A.P. 64, 60, 56, 52, .....is zero?
(a) 16th
(b) 17th
(c) 14th
(d) 15th
Answer: (b) 17th

Question. The nth term of an A.P. is (3n + 5). Its 7th term is –
(a) 26
(b) (3n–2)
(c) 3n + 12
(d) cannot be determined
Answer: (a) 26

Question. The sides of a right angle triangle are in A.P. The ratio of side is –
(a) 1 : 2 : 3
(b) 2 : 3 : 4
(c) 3 : 4 : 5
(d) 5 : 8 : 3
Answer: (c) 3 : 4 : 5

Question. The sum of 1, 3, 5, 7, 9, ....... upto 20 terms is–
(a) 400
(b) 563
(c) 472
(d) 264
Answer: (a) 400

Question. The sum of the series 5 + 13 + 21 + ... + 181 is –
(a) 2139
(b) 2476
(c) 2219
(d) 2337
Answer: (a) 2139

Question. The sum of all odd numbers between 100 and 200 is –
(a) 6200
(b) 6500
(c) 7500
(d) 3750
Answer: (c) 7500

Question. The sum of all positive integral multiples of 5 less than 100 is –
(a) 950
(b) 1230
(c) 760
(d) 875
Answer: (a) 950

Question. The sum of all even natural numbers less than 100 is –
(a) 2450
(b) 2272
(c) 2352
(d) 2468
Answer: (a) 2450

Question. Arithmetic mean between 14 and 18 is –
(a) 16
(b) 15
(c) 17
(d) 32
Answer: (a) 16

Question. If 4, \( A_1, A_2, A_3 \), 28 are in A.P., then the value of \( A_3 \) is –
(a) 23
(b) 22
(c) 19
(d) cannot be determined
Answer: (b) 22

Question. How many terms of the A.P. 3, 6, 9, 12, 15, ..... must be taken to make the sum 108 ?
(a) 6
(b) 7
(c) 8
(d) 36
Answer: (c) 8

Question. The 6th and 8th terms of an A.P. are 12 and 22 respectively, Its 2nd term is –
(a) 8
(b) –8
(c) 6
(d) –3
Answer: (b) –8

Question. In an AP, then sum of first n terms is \( \left( \frac{3n^2}{2} + \frac{5n}{2} \right) \). Find its 25th term.
(a) 924
(b) 76
(c) 1924
(d) 1848
Answer: (b) 76

Question. 200 logs are stocked in such a way that there are 20 logs in the bottom row, 19 in the next row, 18 in the next row and so on. In how many row 200 logs are placed and how many logs are there in the top row ?
(a) 19, 5
(b) 16, 5
(c) 10, 20
(d) 20, 7
Answer: (b) 16, 5