CBSE Class 10 Mathematics Arithmetic Progression MCQs Set M

Question. If the sides of a right triangle are in A.P., then the ratio of its smallest side to the greatest side is :-
(a) 3 : 4
(b) 3 : 5
(c) 4 : 5
(d) None of the options
Answer: (c) 4 : 5

Question. Given that n A.M.'s are inserted between two sets of numbers a, 2b and 2a, b, where a, b \(\in\) R. If the mth means in the two cases are same then ratio a : b is equal to :-
(a) n : (n – m + 1)
(b) (n – m + 1) : m
(c) (n – m + 1) : n
(d) m : (n – m + 1)
Answer: (d) m : (n – m + 1)

Question. The next term of the sequence 9, 16, 27, 42, ......... is :-
(a) 53
(b) 61
(c) 57
(d) None of the options
Answer: (b) 61

Question. Sum of first n terms of an A.P. is an(n – 1). The sum of squares of these terms is :-
(a) \(\frac{a^2}{6} n(n – 1) (2n – 1)\)
(b) \(\frac{2a^2}{3} n(n + 1) (2n + 1)\)
(c) \(a^2n^2(n – 1)^2\)
(d) \(\frac{2a^2}{3} n(n – 1) (2n – 1)\)
Answer: (d) \(\frac{2a^2}{3} n(n – 1) (2n – 1)\)

Question. The nth term of the series \(1 + \frac{1+3}{3} + \frac{1+3+5}{5} + ...\) is :-
(a) \(\frac{2}{n(n+1)}\)
(b) \(\frac{n^2}{2n-1}\)
(c) \(n^2\)
(d) None of the options
Answer: (b) \(\frac{n^2}{2n-1}\)

Question. If a, b, c, d, e, f are A.M.s between 2 and 12, then a + b + c + d + e + f is equal to :-
(a) 14
(b) 84
(c) 42
(d) None of the options
Answer: (c) 42

Question. The sum of all numbers from 1 to 1000 which are neither divisible by 2 nor by 5 is :-
(a) 200000
(b) 500500
(c) 250000
(d) None of the options
Answer: (a) 200000

Question. The nth term of the sequence 2, 5, 11, 20, 32, ............. is :-
(a) \(\frac{3n^2 + 3n - 4}{2}\)
(b) \(\frac{3n^2 - 3n + 4}{2}\)
(c) \(\frac{3n^2 + 3n + 4}{2}\)
(d) None of the options
Answer: (b) \(\frac{3n^2 - 3n + 4}{2}\)

Question. For the A.P. a + (a + d) + (a + 2d) + .... + \(\ell\) of n terms :-
(a) \(S_n = \frac{n}{2} (a + \ell)\)
(b) \(S_n = \frac{n}{2} \{2a + (n–1) d\}\)
(c) \(S_n = \frac{n}{2} \{2\ell – (n–1) d\}\)
(d) \(S_n = \frac{(\ell - a + d)(a + \ell)}{2d}\)
Answer: (a), (b), (c), (d)

Question. If a, b, c are in A.P. then :-
(a) the equation \((b – c) x^2 + (c – a) x + (a – b) = 0\), \(b \neq c\) has equal roots
(b) \(a^2, b^2, c^2\) are in A.P.
(c) \(\lambda a + \mu, \lambda b + \mu, \lambda c + \mu\) are in A.P., \(\lambda, \mu \in R\)
(d) None of the options
Answer: (a), (c)

Question. The sums of first n terms of two A.P.'s are in the ratio (3n + 8) : (7n + 15). The ratio of their 12th terms is :-
(a) \(\frac{4}{9}\)
(b) \(\frac{7}{16}\)
(c) \(\frac{3}{7}\)
(d) None of the options
Answer: (b) \(\frac{7}{16}\)

Question. The sum of n terms of a series is \(An^2 + Bn\), then the nth term is :-
(a) A(2n – 1) – B
(b) A(1 – 2n) + B
(c) A(1 – 2n) – B
(d) A(2n – 1) + B
Answer: (d) A(2n – 1) + B

Question. In an A.P. sum of first p terms is equal to the sum of first q terms. Sum of it's first p + q terms is :-
(a) – (p + q)
(b) p + q
(c) 0
(d) None of the options
Answer: (c) 0

Question. \(2, \sqrt{6} , 4.5\) are the following terms of an A.P.
(a) 101st, 207th, 309th
(b) 101st, 201st, 301st
(c) 2nd, 6th, 9th
(d) None of the options
Answer: (d) None of the options

Question. The sum of 40 A.M's between two numbers is 120. The sum of 50 A.M's between them is equal to :-
(a) 130
(b) 160
(c) 150
(d) None of the options
Answer: (c) 150

Question. The sum of first n terms of an A.P. whose last term is \(\ell\) and common difference is d is :-
(a) \(\frac{n}{2} [2\ell + (n – 1) d]\)
(b) \(\frac{n}{2} [2\ell – (n – 1) d]\)
(c) \(\frac{n}{2} [\ell + (n – 1) d]\)
(d) \(\frac{n}{2} [\ell – (n – 1) d]\)
Answer: (b) \(\frac{n}{2} [2\ell – (n – 1) d]\)

Question. In an A.P., sum of first n terms is \(2n^2 + 3n\), it's common difference is :-
(a) 4
(b) 3
(c) 2
(d) 6
Answer: (a) 4

Question. The number of terms common to the arithmetic progressions 3, 7, 11, ......., 407 and 2, 9, 16....., 709 is :-
(a) 51
(b) 14
(c) 21
(d) 28
Answer: (b) 14

Question. If \(\frac{a^n + b^n}{a^{n-1} + b^{n-1}}\) be the arithmetic mean between a and b, then the value of n is :-
(a) 1
(b) 0
(c) – \(\frac{1}{2}\)
(d) –1
Answer: (a) 1

Question. If the sum of first n terms of an A.P. is \(Pn + Qn^2\) where P and Q are constants, then common difference of A.P. will be :-
(a) P + Q
(b) P – Q
(c) 2P
(d) 2Q
Answer: (d) 2Q

Question. If x, y, z are in A.P., then (x + y – z) (y + z – x) is equal to :-
(a) \(8xy + 3y^2 – 4x^2\)
(b) \(8xy – 3y^2 – 4x^2\)
(c) \(8xy – 3x^2 + 4y^2\)
(d) \(8xy – 3y^2 + 4x^2\)
Answer: (b) \(8xy – 3y^2 – 4x^2\)

Question. If an A.P., \(S_m : S_n :: m^2 : n^2\). The ratio of the pth term to qth term is :-
(a) \(\frac{p-1}{q-1}\)
(b) \(\frac{p}{q}\)
(c) \(\frac{2p-1}{2q-1}\)
(d) None of the options
Answer: (c) \(\frac{2p-1}{2q-1}\)

Question. If x, y, z are in A.P., then (x + 2y – z) (x + z – y) (z + 2y – x) is equal to :-
(a) xyz
(b) 2xyz
(c) 4xyz
(d) None of the options
Answer: (c) 4xyz

Question. The value of n, for which \(\frac{a^{n+1} + b^{n+1}}{a^n + b^n}\) is the A.M. between a and b is :-
(a) 0
(b) 1
(c) – \(\frac{1}{2}\)
(d) –1
Answer: (a) 0

Question. For an A.P., \(\frac{S_{kn}}{S_n}\) is independent of n. The value of \(\frac{d}{a}\) for this A.P. is :-
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. If S denotes the sum of first n terms of the A.P. a + (a + d) + (a + 2d) + ....... whose nth term is \(\ell\), then the common 'd' of the A.P. is :-
(a) \(\frac{\ell - a}{n}\)
(b) \(\frac{\ell^2 - a^2}{2S - a + \ell}\)
(c) \(\frac{\ell^2 - a^2}{2S - (a + \ell)}\)
(d) None of the options
Answer: (c) \(\frac{\ell^2 - a^2}{2S - (a + \ell)}\)

Question. For the A.P. x + (x + 1) + (x + 2) + ...... + y
(a) C. D. is 1
(b) Numer of terms is y – x + 1
(c) Sum of the series is \(\frac{y-x+1}{2}(x + y)\)
(d) None of the options
Answer: (a), (b), (c)

Question. If the angles A < B < C of a triangle are in A.P. then :-
(a) \(c^2 = a^2 + b^2 – ab\)
(b) \(b^2 = a^2 + c^2 – ac\)
(c) \(c^2 = a^2 + b^2\)
(d) None of the options
Answer: (c) \(c^2 = a^2 + b^2\)

Question. The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the same progression. Then which element of the series should necessarily be equal to zero ?
(a) 1st
(b) 9th
(c) 12th
(d) None of the options
Answer: (c) 12th

Question. A person pays Rs. 975 in monthly instalments, each monthly instalment being less than the former by Rs. 5. The amount of the first instalment is Rs. 100. In what tune, will the entire amount be paid ?
(a) 12 months
(b) 26 months
(c) 15 months
(d) 18 months
Answer: (c) 15 months

Question. Let \(S_n\) denote the sum of the first ‘n’ terms of an A.P. \(S_{2n}= 3S_n\). Then, the ratio \(S_{3n}/S_n\) is equal to
(a) 4
(b) 6
(c) 8
(d) 10
Answer: (b) 6

Question. If the nth term of an A.P. is 4n + 1, then the common difference is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (b) 4

Question. 30 trees are planted in a straight line at intervals of 5 m. To water them, the gardener needs to bring water for each tree, separately from a well, which is 10 m from the first tree in line with the trees. How far will he have to walk in order to water all the trees beginnings with the first tree ? Assume that he starts from the well.
(a) 4785 m
(b) 4795 m
(c) 4800 m
(d) None of the options​​​​​​​
Answer: (b) 4795 m

Question. If \(\frac{3+5+7+.........+n \text{ terms}}{5+8+11+......+10 \text{ terms}} = 7\), then the value of n is
(a) 35
(b) 36
(c) 37
(d) 40
Answer: (a) 35

Question. If the sum of first n natural numbers is one-fifth of the sum of their squares, then n is
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (c) 7

Question. The sum of 12 terms of an A.P. whose first term is 4, is 256. What is the last terms ?
(a) 35
(b) 36
(c) 37
(d) 116/3
Answer: (c) 37

Question. Find the sum of all natural numbers not exceeding 1000, which are divisible by 4 but not by 8.
(a) 62500
(b) 62800
(c) 64000
(d) 65600
Answer: (a) 62500

Question. I open a book store with a number of books. On the first day, I sell 1 book; on the second day, I sell 2 books; on the third day, I sell 3 books and so on. At the end of the month (30 days). I realise that I sold the same number of books with which I started. Find the number of books in the beginning.
(a) 365
(b) 420
(c) 465
(d) 501
Answer: (c) 465

Question. There are two arithmetic progressions, \(A_1\) and \(A_2\), whose first terms are 3 and 5 respectively and whose common differences are 6 and 8 respectively. How many terms of the series are common in the first n terms of \(A_1\) and \(A_2\), if the sum of the nth terms of \(A_1\) and \(A_2\) is equal to 6,000?
(a) 103
(b) 107
(c) 109
(d) 113
Answer: (b) 107

Question. A club consists of members whose ages are in AP, the common difference being 3 months. If the youngest member of the club is just 7 years old and the sum of the ages of all the members is 250 year, then the number of members in the club are
(a) 15
(b) 20
(c) 25
(d) 30
Answer: (c) 25

Question. How many terms are there is an AP whose first and fifth terms are –14 and 2 respectively and the sum of terms is 40?
(a) 15
(b) 10
(c) 5
(d) 20
Answer: (b) 10

Question. A man arranged to pay off a debt of Rs.3600 in 40 annual instalments which form an Arithmetical Progression. When 30 of the instalments are paid, he dies leaving one third of the debt unpaid. Find the value of the 1 instalment is
(a) Rs.55
(b) Rs.53
(c) Rs.51
(d) Rs.49
Answer: (c) Rs.51

Question. Let \(a_1, a_2, .......a_{19}\) be the first 19 terms of an arithmetic progression where \(a_1+a_8+a_{12}+a_{19}=224\). The sum \(a_1 + a_2 + a_3 +...+a_{19}\) is equal to
(a) 896
(b) 969
(c) 1064
(d) 1120
Answer: (c) 1064

Question. How many multiples of 7 are there between 33 and 329 ?
(a) 43
(b) 35
(c) 329
(d) 77
Answer: (a) 43

Question. The infinite sum \(1 + \frac{4}{7} + \frac{9}{7^2} + \frac{16}{7^3} + \frac{25}{7^4} + ....\) equals
(a) \(\frac{27}{14}\)
(b) \(\frac{21}{13}\)
(c) \(\frac{49}{27}\)
(d) \(\frac{256}{147}\)
Answer: (c) \(\frac{49}{27}\)