CBSE Class 10 Mathematics Arithmetic Progression MCQs Set 15

Multiple Choice Questions

Question. Which of the following is not an A.P.?
(a) \( -1.2, 0.8, 2.8, .... \)
(b) \( 3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}, ... \)
(c) \( \frac{4}{3}, \frac{7}{3}, \frac{9}{3}, \frac{12}{3}, ... \)
(d) \( \frac{-1}{5}, \frac{-2}{5}, \frac{-3}{5}, ... \)
Answer: (c) \( \frac{4}{3}, \frac{7}{3}, \frac{9}{3}, \frac{12}{3}, ... \)
 

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) 3.5

Question. The list of numbers –10, –6, –2, 2, ... is:
(a) an AP with \( d = -16 \)
(b) an AP with \( d = 4 \)
(c) an AP with \( d = -4 \)
(d) not an AP
Answer: (b) an AP with \( d = 4 \)

Question. The first term of an A.P. is 5 and the last term is 45. If the sum of all the terms is 400, the number of terms is
(a) 20
(b) 8
(c) 10
(d) 16
Answer: (d) 16

Question. The common difference of the A.P.
\( \frac{1}{p}, \frac{1-p}{p}, \frac{1-2p}{p}, ......... \) is
(a) \( 1 \)
(b) \( \frac{1}{p} \)
(c) \( -1 \)
(d) \( -\frac{1}{p} \) 
Answer: (c) -1

Question. The \( n^{th} \) term of the A.P. \( a, 3a, 5a, ....... \) is
(a) \( na \)
(b) \( (2n - 1)a \)
(c) \( (2n + 1)a \)
(d) \( 2na \)
Answer: (b) \( (2n - 1)a \)

Question. The \( 11^{th} \) term of the AP: \( -5, \frac{-5}{2}, 0, \frac{5}{2}, ... \) is
(a) -20
(b) 20
(c) -30
(d) 30 
Answer: (b) 20

Question. The first four terms of an AP, whose first term is -2 and the common difference is -2, are:
(a) -2, 0, 2, 4
(b) -2, 4, -8, 16
(c) -2, -4, -6, -8
(d) -2, -4, -8, -16 
Answer: (c) -2, -4, -6, -8

Question. The \( 21^{st} \) term of the AP whose first two terms are -3 and 4 is:
(a) 17
(b) 137
(c) 143
(d) -143 
Answer: (b) 137

Question. Which term of the AP: 21, 42, 63, 84, ... is 210?
(a) \( 9^{th} \)
(b) \( 10^{th} \)
(c) \( 11^{th} \)
(d) \( 12^{th} \)
Answer: (b) \( 10^{th} \)

Question. The value of \( x \) for which \( 2x \), \( (x + 10) \) and \( (3x + 2) \) are the three consecutive terms of an AP, is
(a) 6
(b) -6
(c) 18
(d) -18 
Answer: (a) 6

Question. The first term of an AP is \( p \) and the common difference is \( q \), then its 10th term is
(a) \( q + 9p \)
(b) \( p - 9q \)
(c) \( p + 9q \)
(d) \( 2p + 9q \) 
Answer: (c) \( p + 9q \)

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) 25

Question. Two APs have the same common difference. The first term of one of these is -1 and that of the other is -8. Then the difference between their \( 4^{th} \) terms is:
(a) -1
(b) -8
(c) 7
(d) -9 
Answer: (c) 7

Question. The famous mathematician associated with finding the sum of the first 100 natural numbers is:
(a) Pythagoras
(b) Newton
(c) Gauss
(d) Euclid
Answer: (c) Gauss

Question. If \( k \), \( 2k - 1 \) and \( 2k + 1 \) are three consecutive terms of an AP, then the value of \( k \) is:
(a) 2
(b) 3
(c) -3
(d) 5
Answer: (b) 3

Question. If the first term of an AP is -5 and the common difference is 2, then the sum of the first 6 terms is
(a) 0
(b) 5
(c) 6
(d) 15
Answer: (a) 0

Question. The \( 11^{th} \) term of the AP: \( \sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, .... \) is:
(a) \( 17\sqrt{2} \)
(b) \( 19\sqrt{2} \)
(c) \( 21\sqrt{2} \)
(d) \( 23\sqrt{2} \)
Answer: (c) \( 21\sqrt{2} \)

Question. The sum of the first 16 terms of the AP: 10, 6, 2, ... is:
(a) -320
(b) 320
(c) -352
(d) -400
Answer: (a) -320

Question. In an AP if \( a = 1 \), \( a_n = 20 \) and \( S_n = 399 \), then \( n \) is:
(a) 19
(b) 21
(c) 38
(d) 42
Answer: (c) 38
 

Fill in the Blanks

Question. Fill the two blanks in the sequence 2, ....., 26, ..... so that the sequence forms an A.P.
Answer: 14, 38 

Question. The sum of first 16 terms of the AP 5, 8, 11, 14, ...... is ............................... .
Answer: 440
Explanation: Here first term = 5
Common difference = \( 8 - 5 = 3 \)
Number of terms = 16
\( S_n = \frac{n}{2}[2a + (n - 1)d] \)
\( S_{16} = \frac{16}{2}[2 \times 5 + (16 - 1) \times 3] \)
\( = 8[10 + 45] \)
\( = 8 \times 55 = 440 \)

Question. The common difference of an A.P. 6, then \( a_{15} - a_{11} \) .................................. .
Answer: 24
Explanation: Let \( a \) be the first term and \( d \) be the common difference.
\( n^{th} \) term = \( a_n = a + (n - 1)d \)
Now, \( a_{15} = a + (15 - 1)d = a + 14d \)
\( a_{11} = a + (11 - 1)d = a + 10d \)
\( a_{15} - a_{11} = (a + 14d) - (a + 10d) = 4d \)
As \( d = 6 \)
\( a_{15} - a_{11} = 4 \times 6 = 24 \)

Question. If 4/5, a, 2 are three consecutive terms of an AP then the value of a is ............................... .
Answer: \( \frac{7}{5} \)
Explanation: Given \( \frac{4}{5}, a, 2 \) are in AP
Then, \( a - \frac{4}{5} = 2 - a \)

\( \Rightarrow \) \( 2a = 2 + \frac{4}{5} \)

\( \Rightarrow \) \( 2a = \frac{14}{5} \)

\( \Rightarrow \) \( a = \frac{7}{5} \)

Question. If \( 4, x_1, x_2, x_3, 28 \) are in AP then \( x_3 \) = ...................... .
Answer: 22
Explanation: Given, \( 4, x_1, x_2, x_3, 28 \) are in AP
Let \( d \) be the common difference
Now, first term, \( a = 4 \)
and fifth term, \( a_5 = 28 \)
\( a_5 = a + (5 - 1)d = a + 4d \)

\( \Rightarrow \) \( 28 = 4 + 4d \)

\( \Rightarrow \) \( 4d = 24 \)

\( \Rightarrow \) \( d = 6 \)
\( x_3 = a + 3d = 4 + 3 \times 6 = 22 \)

Question. If \( S_n = 5n^2 + 3n \), then \( n^{th} \) term is ........................... .
Answer: \( 10n - 2 \)
Explanation:
\( a_n = S_n - S_{n-1} \)
\( = 5n^2 + 3n - [5(n - 1)^2 + 3(n - 1)] \)
\( = 5n^2 + 3n - [5n^2 + 5 - 10n + 3n - 3] \)
\( = 10n - 2 \)

Question. Find the \( 16^{th} \) term of the AP: 2, 7, 12, 17, ....... .
Answer: 77
Explanation: Here, \( a = 2 \)
\( d = 7 - 2 = 12 - 7 = 5 \)
\( a_{16} = a + (16 - 1)d \)
\( = 2 + 15 \times 5 \)
\( = 2 + 75 = 77 \)

Question. The number of terms of AP: 18, 16, 14, ... that make the sum zero, is ................
Answer: 19
Explanation: Let \( n \) terms of the given AP make the sum zero.
Then,
\( \frac{n}{2}[18 \times 2 + (n - 1)(-2)] = 0 \)

\( \Rightarrow \) \( 36 - 2(n - 1) = 0 \)

\( \Rightarrow \) \( 36 - 2n + 2 = 0 \)

\( \Rightarrow \) \( 2n = 38 \)

\( \Rightarrow \) \( n = 19 \)

Question. Second term of the AP if its \( S_n = n^2 + 2n \) is .....................
Answer: 5
Explanation:
Here, \( a_2 = S_2 - S_1 = (2^2 + 2 \times 2) - (1^2 + 2 \times 1) \)
\( = (4 + 4) - (1 + 2) \)
\( = 5 \)

Question. \( 10^{th} \) term from end of AP: 4, 9, 14, ...., 254 is ............................... .
Answer: 209
Explanation: The given AP in reverse form is:
254, 249, 244, ...., 14, 9, 4.
Here, \( a = 254 \), \( d = -5 \)
So, \( a_{10} = 254 + 9(-5) \)
\( = 254 - 45 = 209 \)