CBSE Class 10 Arithmetic Progressions Sure Shot Questions Set M

Sequence, series & progression

\( S_n = T_1 + T_2 + T_3 + \dots + T_{n-2} + T_{n-1} + T_n \)

\( S_{n-1} = T_1 + T_2 + T_3 + \dots + T_{n-2} + T_{n-1} \)

\( S_n - S_{n-1} = T_n \) OR \( T_n = S_n - S_{n-1} \)

• Sequence : A sequence is an ordered arrangement of numbers according to a given rule.
• Terms of a sequence : The terms of a sequence in successive order is denoted by \( T_n \) or \( a_n \). The nth term \( T_n \) is called the general term of the sequence.
• Series: The sum of terms of a sequence is called the series of the corresponding sequence. \( T_1 + T_2 + T_3 + \dots \) is an infinite series, where as \( T_1 + T_2 + T_3 + \dots + T_{n-1} + T_n \) is a finite series of n terms. Usually the series of finite number of n terms is denoted by \( S_n \).
• Progression : The sequence that follows a certain pattern is called a progression. Thus, the sequence 2, 3, 5, 7, 11,.... is not a progression.

Arithmetic progressions

An arithmetic progression is that list of numbers in which the first term is given and each term, other than the first term is obtained by adding a fixed number 'd' to the preceding term.

The fixed term 'd' is known as the common difference of the arithmetic progression. It's value can be positive, negative or zero. The first term is denoted by 'a' or '\( a_1 \)' and the last term by '\( l \)'.

• Symbolical form : Let us denote the first term of an AP by \( a_1 \), second term by \( a_2 \), .... nth term by \( a_n \) and the common difference by d. Then the AP becomes \( a_1, a_2, a_3, \dots, a_n \). So, \( a_2 - a_1 = a_3 - a_2 = \dots = a_n - a_{n-1} = d \).
• General form : In general form, an arithmetic progression with first term 'a' and common difference 'd' can be represented as follows :
  \( a, a + d, a + 2d, a + 3d, a + 4d, \dots \)
• Finite AP : An AP in which there are only a finite number of terms is called a finite AP. It may be noted that each such AP has a last term.
• General term of an arithmetic progression
  The nth term of an arithmetic progression is
  \( a_n = a + (n - 1)d \)
  Where, a is the first term of arithmetic progression, and d is the common difference of arithmetic progression.
• \( r^{th} \) term of a finite arithmetic progression from the end
  Let there be an arithmetic progression with first term a and common difference d. If there are n terms in the arithmetic progression, then
  \( r^{th} \) term from the end = \( a + (n - r)d \)
  Also, if \( l \) is the last term of the arithmetic progression then \( r^{th} \) term from the end is the \( r^{th} \) term of an arithmetic progression whose first term is \( l \) and common difference is \( -d \).
  \( r^{th} \) term from the end = \( l + (r - 1)(-d) \)
• In the formula \( S_n = \frac{n}{2} [2a + (n - 1)d] \), there are four quantities viz. \( S_n \), a, n and d. If any three of these are known, the fourth can be determined. Sometimes, two of these quantities are given. In such a case, remaining two quantities are provided by some other relation.
• If the sum \( S_n \) of n terms of a sequence is given, then nth term \( a_n \) of the sequence can be determined by using the following formula : \( a_n = S_n - S_{n-1} \)
  i.e., the nth term of an AP is the difference of the sum to first n terms and the sum to first (n – 1) terms of it.

Selection of terms in an AP

It should be noted that in case of an odd number of terms, the middle term is a and the common difference is d while in case of an even number of terms the middle terms are a – d, a + d and the common difference is 2d.

• If the sum of terms is not given, then select terms as a, a + d, a + 2d,....
• If three numbers a, b, c in order are in AP. Then
  b – a = Common difference = c – b
  \( \Rightarrow b – a = c – b \)
  \( \Rightarrow 2b = a + c \)
  Thus, a,b,c are in AP if and only if \( 2b = a + c \)
• If a,b,c are in AP, then b is known as the arithmetic mean (AM) between a and c.
• If a, x, b are in AP Then, \( 2x = a + b \Rightarrow x = \frac{a+b}{2} \)
  Thus, AM between a and b is \( \frac{a+b}{2} \).

Points to remember

• If a constant is added to or subtracted from each term of an A.P., then the resulting sequence is also an A.P. with the same common difference.
• If each term of a given A.P. is multiplied or divided by a non-zero constant K, then the resulting sequence is also an A.P. with common difference Kd or d/K, where d is the common difference of the given A.P.
• In a finite A.P., the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term.
• A sequence is an A.P. iff it's nth term is a linear expression in n i.e., \( a_n = An + B \), where A, B are constants. In such a case, the coefficient of n is the common difference of the A.P.
• A sequence is an A.P. iff the sum of it's first n terms is of the form \( An^2 + Bn \), where A,B are constants independent of n. In such a case, the common difference is 2 A.
• If the terms of an A.P. are choosen at regular intervals, then they form an A.P.
• The sum of \( 1^{st} \) n odd natural no.s = \( n^2 \)
• The sum of \( 1^{st} \) n even natural no.s = \( n(n+1) \)
• The sum of first n natural numbers i.e. \( 1 + 2 + 3 +....+ n \) is usually written as \( S_n \).
  \( \sum n = \frac{n(n +1)}{2} \)
• The sum of squares of first n natural numbers i.e. \( 1^2 + 2^2 + 3^2 + \dots + n^2 \) is usually written as \( S_{n^2} \).
  \( \sum n^2 = \frac{n(n +1)(2n +1)}{6} \)
• The sum of cubes of first n natural numbers i.e. \( 1^3 + 2^3 + 3^3 + \dots + n^3 \) is usually written as \( S_{n^3} \).
  \( \sum n^3 = \left( \frac{n(n +1)}{2} \right)^2 = (S_n)^2 \)
• A sequence of non-zero numbers \( a_1, a_2, a_3, \dots, a_n \) is said to be a geometric sequence or G.P. iff \( \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \dots \)
  i.e. iff \( \frac{a_{n+1}}{a_n} = \text{a constant for all n.} \)
  e.g., 3, 9, 27, 81,....

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
Answer: (b) 3 : 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: (c) (n – m + 1) : n

Question. The next term of the sequence 9, 16, 27, 42, ......... is :-
(a) 53
(b) 61
(c) 57
(d) None
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 \( 1 + \frac{1}{1 + 3} + \frac{1}{1 + 3 + 5} + \dots \) is :-
(a) \( \frac{2}{n(n + 1)} \)
(b) \( \frac{1}{n^2} \)
(c) \( n^2 \)
(d) None
Answer: (b) \( \frac{1}{n^2} \)

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
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 these
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 these
Answer: (b) \( \frac{3n^2 - 3n + 4}{2} \)

Question. For the A.P. \( a + (a + d) + (a + 2d) + \dots + l \) of n terms :-
(a) \( S_n = \frac{n}{2} (a + l) \)
(b) \( S_n = \frac{n}{2} \{2a + (n-1) d\} \)
(c) \( S_n = \frac{n}{2} \{2l - (n-1) d\} \)
(d) \( S_n = \frac{(l - a + d)(a + l)}{2d} \)
Answer: (a, b, c, d) All formulas are correct.

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 these
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) 4/9
(b) 7/16
(c) 3/7
(d) None
Answer: (b) 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
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 these
Answer: (d) None of these

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
Answer: (c) 150

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

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
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
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 \( l \), then the common 'd' of the A.P. is :-
(a) \( \frac{l - a}{n} \)
(b) \( \frac{l^2 - a^2}{2S - a + l} \)
(c) \( \frac{l^2 - a^2}{2S - (a + l)} \)
(d) None of these
Answer: (c) \( \frac{l^2 - a^2}{2S - (a + l)} \)

Question. For the A.P. x + (x + 1) + (x + 2) + ...... + y :-
(a) C. D. is 1
(b) Number of terms is y – x + 1
(c) Sum of the series is \( \frac{y-x+1}{2}(x + y) \)
(d) None of these
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 these
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 these
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 these
Answer: (b) 4795 m

Question. If \( \frac{3+5+7+\dots+n \text{ terms}}{5+8+11+\dots+10 \text{ terms}} = 7 \), 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 term ?
(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 1st instalment is
(a) Rs.55
(b) Rs.53
(c) Rs.51
(d) Rs.49
Answer: (c) Rs.51

Question. Let \( a_1, a_2, \dots, a_{19} \) be the first 19 terms of an arithmetic progression where \( a_1 + a_8 + a_{12} + a_{19} = 224 \). The sum \( a_1 + a_2 + a_3 + \dots + a_{19} \) 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 \( 1 + \frac{4}{7} + \frac{9}{7^2} + \frac{16}{7^3} + \frac{25}{7^4} + \dots \) equals
(a) 27/14
(b) 21/13
(c) 49/27
(d) 256/147
Answer: (c) 49/27