CBSE Class 10 Mathematics Arithmetic Progression MCQs Set 08

Question. If the sum of the series \( 2 + 5 + 8 + 11 + \dots \) is 60100, then the number of terms are
(a) 100
(b) 200
(c) 150
(d) 250
Answer: B
Let, \( S_n = 60100 \)
\( \frac{n}{2} [4 + (n - 1)3] = 60100 \)
\( n(3n + 1) = 120200 \)
\( 3n^2 + n - 120200 = 0 \)
\( (n - 200)(3n + 601) = 0 \)
\( n = 200 \)
\( n = -\frac{601}{3} \) (\( n \) cannot be fraction)

Question. If the common difference of an AP is 5, then what is \( a_{18} - a_{13} \)?
(a) 5
(b) 20
(c) 25
(d) 30
Answer: C
Given, the common difference of AP i.e., \( d = 5 \)
Now, \( a_{18} - a_{13} = [a + (18 - 1)d] - [a + (13 - 1)d] \)
[Since, \( a_n = a + (n - 1)d \)]
\( = a + 17 \times 5 - a - 12 \times 5 \)
\( = 85 - 60 = 25 \)

Question. What is the common difference of four terms in A.P. such that the ratio of the product of the first fourth term to that of the second and third term is 2:3 and the sum of all four terms is 20?
(a) 3
(b) 1
(c) 4
(d) 2
Answer: D
Take the four terms as \( a - 3x, a - x, a + x, a + 3x \)
The sum \( = 4a = 20 \)
\( a = 5 \)
Also, \( 3(a^2 - (3x)^2) = 2(a^2 - x^2) \)
\( x = 1 \)
However, the common difference is \( 2x \) and not \( x \)
When, \( x = 1, d = 2x = 2 \)

Question. There are 60 terms is an A.P. of which the first term is 8 and the last term is 185. The \( 31^{st} \) term is
(a) 56
(b) 94
(c) 85
(d) 98
Answer: D
Let \( d \) be the common difference;
Then \( 60^{th} \) term, \( = 8 + 59d = 185 \)
\( 59d = 177 \)
\( d = 3 \)
\( 31^{th} \) term \( = 8 + 30 \times 3 = 98 \)

Question. The first and last term of an A.P. are \( a \) and \( l \), respectively. If \( S \) is the sum of all the terms of the A.P. and the common difference is \( \frac{l^2 - a^2}{k - (l + a)} \), then \( k \) is equal to
(a) \( S \)
(b) \( 2S \)
(c) \( 3S \)
(d) None of these
Answer: B
We have, \( S = \frac{n}{2}(a + l) \)
\( \frac{2S}{a + l} = n \dots (1) \)
Also, \( l = a + (n - 1)d \)
\( d = \frac{l - a}{n - 1} = \frac{l - a}{\frac{2S}{a + l} - 1} \)
\( = \frac{l^2 - a^2}{2S - (l + a)} \)
\( k = 2S \)

Question. The sum of 11 terms of an A.P. whose middle term is 30, is
(a) 320
(b) 330
(c) 340
(d) 350
Answer: B
Middle term \( = a_6 = a + (n - 1)d \)
\( a_6 = a + 5d \)
Also, \( a_6 = 30 \)
Hence, \( 30 = a + 5d \)
\( a = 30 - 5d \)
Putting value of \( a \) in eqn (1):
\( S_{11} = \frac{11}{2} [2(30 - 5d) + 10d] \)
\( S_{11} = \frac{11}{2} [60 - 10d + 10d] \)
\( S_{11} = 11 \times 30 = 330 \)

Question. There are four arithmetic means between 2 and -18. The means are
(a) -4, -7, -10, -13
(b) 1, -4, -7, -10
(c) -2, -5, -9, -13
(d) -2, -6, -10, -14
Answer: D
Let the means be \( X_1, X_2, X_3, X_4 \) and the common difference be \( b \); then \( 2, X_1, X_2, X_3, X_4, -18 \) are in A.P.;
\( -18 = 2 + 5b \)
\( 5b = -20 \)
\( b = -4 \)
Hence, \( X_1 = 2 + b = 2 + (-4) = -2 \);
\( X_2 = 2 + 2b = 2 - 8 = -6 \)
\( X_3 = 2 + 3b = 2 - 12 = -10 \)
\( X_4 = 2 + 4b = 2 - 16 = -14 \)
The required means are -2, -6, -10, -14.

Question. If the \( n^{th} \) term of an A.P. is given by \( a_n = 5n - 3 \), then the sum of first 10 terms if
(a) 225
(b) 245
(c) 255
(d) 270
Answer: B
Putting, \( n = 1, 10 \)
we get, \( a = 2 \)
\( l = 47 \)
\( S_{10} = \frac{10}{2} (2 + 47) = 5 \times 49 = 245 \)

Question. Find the sum of the series \( 1 + (1+2) + (1+2+3) + (1+2+3+4) + \dots + (1+2+3+\dots+20) \)
(a) 1470
(b) 1540
(c) 1610
(d) 1370
Answer: B
Let \( S = 1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + \dots + (1 + 2 + 3 + \dots + 20) \)
\( = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 + 105 + 120 + 136 + 153 + 171 + 190 + 210 \)
\( = 1540 \)
[Since, \( 1+2+3+\dots+n = \frac{n(n+1)}{2} \)]

Question. An AP starts with a positive fraction and every alternate term is an integer. If the sum of the first 11 terms is 33, then the fourth term is
(a) 2
(b) 3
(c) 5
(d) 6
Answer: A
Given, \( S_{11} = 33 \)
\( \frac{11}{2} [2a + 10d] = 33 \Rightarrow a + 5d = 3 \)
i.e., \( a_6 = 3 \Rightarrow a_4 = 2 \)
[Since, Alternate terms are integers and the given sum is possible]

Question. Five distinct positive integers are in a arithmetic progression with a positive common difference. If their sum is 10020, then the smallest possible value of the last term is
(a) 2002
(b) 2004
(c) 2006
(d) 2007
Answer: C
Let the five integers be \( a - 2d, a - d, a, a + d, a + 2d \).
Then, we have,
\( (a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 10020 \)
\( 5a = 10020 \Rightarrow a = 2004 \)
Now, as smallest possible value of \( d \) is 1.
Hence, the smallest possible value of \( a + 2d \) is \( 2004 + 2 = 2006 \)

Question. In an AP, if \( a = 3.5 \), \( d = 0 \) and \( n = 101 \), then \( a_n \) will be
(a) 0
(b) 3.5
(c) 103.5
(d) 104.5
Answer: B
For an AP, \( a_n = a + (n - 1)d \)
\( = 3.5 + (101 - 1) \times 0 \)
[by given conditions]
\( = 3.5 \)

Question. The number of common terms to the two sequences 17, 21, 25, \dots, 417 and 16, 21, 26, \dots, 466 is
(a) 19
(b) 20
(c) 21
(d) 91
Answer: B
Common terms will be 21, 41, 61, \dots
\( 21 + (n - 1)20 \leq 417 \)
\( n \leq 20.8 \)
\( n = 20 \)

Question. If the sum of the first \( 2n \) terms of 2, 5, 8, \dots is equal to the sum of the first \( n \) terms of 57, 59, 61, \dots, then \( n \) is equal to
(a) 10
(b) 12
(c) 11
(d) 13
Answer: C
Given, \( \frac{2n}{2} \{2 \cdot 2 + (2n - 1)3\} = \frac{n}{2} \{2 \cdot 57 + (n - 1)2\} \)
or \( 2(6n + 1) = 112 + 2n \)
or \( 10n = 110 \)
\( n = 11 \)

Question. Let \( T_r \) be the \( r^{th} \) term if an A.P. for \( r = 1, 2, 3, \dots \). If for some positive integers \( m, n \), we have \( T_m = \frac{1}{n} \) and \( T_n = \frac{1}{m} \), then \( T_{mn} \) equals
(a) \( \frac{1}{mn} \)
(b) \( \frac{1}{m} + \frac{1}{n} \)
(c) 1
(d) 0
Answer: C
Let, \( T_m = a + (m - 1)d = \frac{1}{n} \dots (1) \)
and \( T_n = a + (n - 1)d = \frac{1}{m} \dots (2) \)
On subtracting Eq. (2) from Eq. (1), we get
\( (m - n)d = \frac{1}{n} - \frac{1}{m} = \frac{m - n}{mn} \)
\( d = \frac{1}{mn} \)
Again, \( T_{mn} = a + (mn - 1)d \)
\( = a + (mn - n + n - 1)d \)
\( = a + (n - 1)d + (mn - n)d \)
\( = T_n + n(m - 1) \frac{1}{mn} \)
\( = \frac{1}{m} + \frac{(m - 1)}{m} = 1 \)

Question. In an AP, if \( d = -4, n = 7 \) and \( a_n = 4 \), then \( a \) is equal to
(a) 6
(b) 7
(c) 20
(d) 28
Answer: D
In an AP, \( a_n = a + (n - 1)d \)
\( 4 = a + (7 - 1)(-4) \) [by given conditions]
\( 4 = a + (-6 \times 4) \)
\( 4 + 24 = a \)
\( a = 28 \)

FILL IN THE BLANK

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. Number of rows in the flower bed is ...........
Answer: \( n = 10 \)

Question. In the sequence 5, 6, 7, 8 difference between two consecutive terms is ..........
Answer: 1

Question. 4, 10, 16, 22, .........., ..........
Answer: 28, 34

Question. In an AP, the letter d is generally used to denote the ..........
Answer: common difference

Question. 1, -1, -3, -5, .........., ..........
Answer: -7, -9

Question. The sum of n terms of an A.P. is \( 4n^2 - n \). The common difference = .......... .
Answer: 8

Question. 11th term from last term of an A.P. 10, 7, 4 .........., -62, is ..........
Answer: -32

Question. If a and d are respectively the first term and the common difference of an AP, \( a + 10d \), denotes the .......... term of the AP.
Answer: eleventh

Question. If l and d are respectively the last term and the common difference of an AP, then \( l - 9d \) denotes the .......... term of the AP.
Answer: tenth

Question. If 1, 4, 9 form a sequence, the next term is ..........
Answer: 25

TRUE/FALSE

Question. The general form of an A.P. is \( a, a + d, a + 2d, a + 3d, \dots \)
Answer: True

Question. If \( a, b, c \) are in AP, then \( b = \frac{a - c}{2} \).
Answer: False

Question. 0, 2, 0, 2, 0 is an AP.
Answer: False

Question. In an Arithmetic progression, the first term is denoted by ‘a’ and ‘d’ is called the common difference.
Answer: True

Question. The list of numbers \( 3, 3^2, 3^3, 3^4, \dots \) forms an AP.
Answer: False

Question. In an AP with first term a and common difference d, the nth term (or the general term) is given by \( a_n = a + (n - 1)d \).
Answer: True

Question. Sequence 1, 4, 9, 16,... is an arithmetic progression.
Answer: False

Question. If l is the last term, the nth term of the AP \( = l + (n - 1)(-d) = l - (n - 1)d \).
Answer: True

Question. The common difference of an AP can be zero or negative.
Answer: True

Question. If l is the last term of the finite AP, say the nth term, then the sum of all terms of the AP is given by: \( S = \frac{n}{2}(a + l) \).
Answer: True

ASSERTION AND REASON

DIRECTION : 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 : Let the positive numbers \( a, b, c \) be in A.P., then \( \frac{1}{bc}, \frac{1}{ac}, \frac{1}{ab} \) are also in A.P.
Reason : If each term of an A.P. is divided by \( abc \), then the resulting sequence is also in A.P.
(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.
Answer: A

Question. Assertion : Common difference of the AP -5, -1, 3, 7, .......... is 4.
Reason : Common difference of the AP \( a, a + d, a + 2d, \dots \) is given by \( d = \text{2nd term} - \text{1st term} \).
(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.
Answer: A

Question. Assertion : Sum of first 10 terms of the arithmetic progression -0.5, -1.0, -1.5, .......... is 27.5
Reason : Sum of n terms of an A.P. is given as \( S_n = \frac{n}{2}[2a + (n - 1)d] \) where \( a = \text{first term}, d = \text{common difference} \).
(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.
Answer: A

Question. Assertion : \( a_n - a_{n-1} \) is not independent of n then the given sequence is an AP.
Reason : Common difference \( d = a_n - a_{n-1} \) is constant or independent of n.
(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.
Answer: D

Question. Assertion : The sum of the series with the nth term \( t_n = (9 - 5n) \) is (465), when no. of terms \( n = 15 \).
Reason : Given series is in A.P. and sum of n terms of an A.P. is \( S_n = \frac{n}{2}[2a + (n - 1)d] \)
(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.
Answer: D

Question. Assertion : Three consecutive terms \( 2k + 1, 3k + 3 \) and \( 5k - 1 \) form an AP than k is equal to 6.
Reason : In an AP \( a, a + d, a + 2d, \dots \), the sum to n terms of the AP be \( S_n = \frac{n}{2}[2a + (n - 1)d] \)
(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.
Answer: B