CBSE Class 10 Arithmetic Progressions Sure Shot Questions Set 04

Arithmetic Progressions

• Sequence/Progression

• Arithmetic Progressions and its \( n \)-th term
• Sum of First \( n \) Terms of an AP

1. Sequence/Progression

Sequence/Progression: A sequence/progression is a succession of numbers or terms formed according to some pattern or rule. Various numbers occurring in a sequence are called terms or elements.

• A sequence with finite number of terms or numbers is called a finite sequence.
• A sequence with infinite number of terms or numbers is called an infinite sequence.

Question. Write first four terms of each of the following sequence, whose general terms are: (i) \( a_n = 3n - 7 \) (ii) \( a_n = (-1)^{n+1} \times 3^n \)
Answer: (i) \( a_n = 3n - 7 \)
\( a_1 = 3 \times 1 - 7 = 3 - 7 = -4, a_2 = 3 \times 2 - 7 = 6 - 7 = -1, \)
\( a_3 = 3 \times 3 - 7 = 9 - 7 = 2 \) and \( a_4 = 3 \times 4 - 7 = 12 - 7 = 5 \)
(ii) \( a_n = (-1)^{n+1} \times 3^n \)
\( a_1 = (-1)^{1+1} \times 3^1 = 3, \)
\( a_2 = (-1)^{2+1} \times 3^2 = (-1)^3 \times 3^2 = -9, \)
\( a_3 = (-1)^4 \times 3^3 = 27 \) and \( a_4 = (-1)^5 \times 3^4 = -81 \)

Question. What is \( 18^{th} \) term of the sequence defined by \( a_n = \frac{n(n-3)}{n+4} \)?
Answer: We have, \( a_n = \frac{n(n-3)}{n+4} \)
Putting \( n = 18 \), we get \( a_{18} = \frac{18 \times (18-3)}{18+4} = \frac{18 \times 15}{22} = \frac{135}{11} \)

Question. If \( a_n = 5n - 4 \) is a sequence, then \( a_{12} \) is
(a) 48
(b) 52
(c) 56
(d) 62
Answer: (c)

Question. If \( a_n = 3n - 2 \), then the value of \( a_7 + a_8 \) is
(a) 39
(b) 41
(c) 47
(d) 53
Answer: (b)

Question. The second term of the sequence defined by \( a_n = 3n + 2 \) is
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (d)

Question. Assertion (A): The arrangement of numbers, i.e., \( -4, 16, -64, 256, -1024, 4096, \dots \) form a sequence.
Reason (R): An arrangement of numbers which are arranged in a definite order according to some rule, is called a sequence.
(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): Sequence \( 1, 5, 9, 13, 17, 21, \dots \) is a finite sequence.
Reason (R): A sequence with finite number of terms or numbers is called a finite sequence.
(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. Write down the first six terms of each of the following sequences, whose general terms are: (a) \( a_n = 5n - 3 \) (b) \( a_n = (-1)^n \cdot 2^{2n} \) (c) \( a_n = \frac{2n+1}{n+2} \) (d) \( a_n = (-1)^{n-1} \cdot n^2 \)
Answer: (a) 2, 7, 12, 17, 22, 27
(b) -4, 16, -64, 256, -1024, 4096
(c) \( 1, \frac{5}{4}, \frac{7}{5}, \frac{3}{2}, \frac{11}{7}, \frac{13}{8} \)
(d) 1, -4, 9, -16, 25, -36

Question. Find the \( 10^{th} \) term of the sequence defined by \( a_n = (-1)^{2n-1} \cdot 5^n \).
Answer: -9765625

Question. Find the difference between the \( 12^{th} \) term and \( 10^{th} \) term of the sequence whose general term is given by \( a_n = 5n - 1 \).
Answer: 10

2. Arithmetic Progression and its \( n^{th} \) Term

An arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number \( d \) to the preceding term, except the first term \( a \). This fixed number is known as common difference of the AP. Common difference of an AP can be negative, positive or zero.

The general form of an AP is \( a, a + d, a + 2d, a + 3d, \dots \)

The \( n^{th} \) term \( a_n \) (or the general term) of an AP is \( a_n = a + (n - 1) d \), where \( a \) is the first term, \( d \) is the common difference and \( n \) is the number of terms. Also, \( d = a_{n + 1} - a_n \).

If \( l \) is the last term of an AP, then \( n^{th} \) term from the end of the AP \( = l + (n - 1)(-d) = l - (n - 1)d \).

Question. In an AP, if \( d = - 4, n = 7, a_n = 4 \), then find the value of \( a \).
Answer: We have \( a_n = 4 \) for \( n = 7 \)
\( a_n = a + (n - 1) d \Rightarrow 4 = a + 6(- 4) \Rightarrow a = 28 \)

Question. Is 0 a term of the AP: 31, 28, 25, \dots? Justify your answer.
Answer: Given AP is 31, 28, 25, \dots
Here, \( a = 31, d = 28 - 31 = - 3 \)
For 0 be a term of this AP, \( 0 = a_n \) for some \( n \Rightarrow 0 = a + (n - 1)d \)
\( \Rightarrow 0 = 31 + (n - 1) (- 3) \Rightarrow 31 - 3n + 3 = 0 \)
\( \Rightarrow - 3n = - 34 \Rightarrow n = \frac{34}{3} = 11\frac{1}{3} \)
which is not possible as \( n \) cannot be a fraction. Therefore, 0 cannot be a term of this AP.

Question. Find the \( 12^{th} \) term from the end of the AP: \( -2, - 4, - 6, \dots, - 100 \).
Answer: Let \( a \) be the first term, \( d \) the common difference and \( l \) the last term of AP.
Here, \( a = -2, d = (-4 + 2) = -2, l = -100 \) and \( n = 12 \)
\( n^{th} \) term from end \( = l - (n - 1) d \)
\( \Rightarrow 12^{th} \) term from end \( = -100 - (12 - 1) (-2) = -100 + 22 = -78 \)

Question. For what value of \( x \): \( 2x, x + 10 \) and \( 3x + 2 \) are in AP?
Answer: Since, given numbers are in AP.
So, \( (x + 10) - 2x = (3x + 2) - (x + 10) \)
\( \Rightarrow -x + 10 = 2x - 8 \) or \( 3x = 18 \) or \( x = 6 \)

Question. Find the \( 25^{th} \) term of the AP: \( -5, -\frac{5}{2}, 0, \frac{5}{2}, \dots \)
Answer: We have, \( a = -5, d = -\frac{5}{2} - (-5) = \frac{5}{2} \)
\( a_n = a + (n - 1) d \)
\( \Rightarrow a_{25} = (- 5) + (25 - 1) \frac{5}{2} = (- 5) + 24\left(\frac{5}{2}\right) = - 5 + 60 = 55 \)

Question. Find the \( 20^{th} \) term from the last term of the AP: 3, 8, 13, \dots, 253.
Answer: Given, last term \( = l = 253 \)
And, common difference \( = d = 8 - 3 = 5 \)
\( \therefore 20^{th} \) term from end \( = l - (n - 1) \times d = 253 - 19 \times 5 = 253 - 95 = 158 \)

Question. Which term of the AP: 3, 8, 13, 18, \dots, is 78?
Answer: Let \( a_n \) be the required term of the AP: 3, 8, 13, 18,\dots
Here, \( a = 3, d = 8 - 3 = 5 \) and \( a_n = 78 \)
Now, \( a_n = a + (n - 1)d \Rightarrow 78 = 3 + (n - 1) \times 5 \Rightarrow 78 - 3 = (n - 1) \times 5 \)
\( \Rightarrow 75 = (n - 1) \times 5 \Rightarrow \frac{75}{5} = n - 1 \Rightarrow 15 = n - 1 \Rightarrow n = 15 + 1 = 16 \)
Hence, \( 16^{th} \) term of given AP is 78.

Question. The sum of the \( 5^{th} \) and \( 7^{th} \) terms of an AP is 52 and the \( 10^{th} \) term is 46. Find the AP.
Answer: Let the first term and the common difference of an AP be ‘\( a \)’ and ‘\( d \)’.
\( \therefore a_5 = a + 4d \) and \( a_7 = a + 6d \)
So, \( a_5 + a_7 = 2a + 10d = 52 \Rightarrow 2a + 10d = 52 \dots(i) \)
Also, \( a_{10} = a + 9d = 46 \Rightarrow a + 9d = 46 \dots(ii) \)
From (i) and (ii), \( d = 5 \) and \( a = 1 \)
So, the AP is as follows 1, 6, 11, 16, 21, \dots

Question. An AP consists of 50 terms of which \( 3^{rd} \) term is 12 and the last term is 106. Find the \( 29^{th} \) term.
Answer: Let \( a \) be the first term and \( d \) be the common difference.
Since, given AP has 50 terms, so \( n = 50 \)
\( \because a_3 = 12 \Rightarrow a + (3 - 1)d = 12 \Rightarrow a + 2d = 12 \dots(i) \)
Also, \( a_{50} = 106 \Rightarrow a + (50 - 1)d = 106 \Rightarrow a + 49d = 106 \dots(ii) \)
Subtracting (i) from (ii), we get \( 47d = 94 \Rightarrow d = \frac{94}{47} = 2 \)
Putting the value of \( d \) in equation (i), we get \( a + 2 \times 2 = 12 \Rightarrow a = 12 - 4 = 8 \)
Here, \( a = 8, d = 2 \)
So, \( 29^{th} \) term of the AP is given by \( a_{29} = a + (29 - 1)d = 8 + 28 \times 2 \Rightarrow a_{29} = 8 + 56 \Rightarrow a_{29} = 64 \)

Question. Find the \( 31^{st} \) term of an AP whose \( 11^{th} \) term is 38 and the \( 16^{th} \) term is 73.
Answer: Let the first term be \( a \) and common difference be \( d \).
Now, given \( a_{11} = 38 \Rightarrow a + (11 - 1)d = 38 \Rightarrow a + 10d = 38 \dots(i) \)
Also, \( a_{16} = 73 \Rightarrow a + (16 - 1)d = 73 \Rightarrow a + 15d = 73 \dots(ii) \)
Now, subtracting (ii) from (i), we get \( -5d = -35 \Rightarrow 5d = 35 \Rightarrow d = \frac{35}{5} = 7 \)
Putting the value of \( d \) in equation (i), we get \( a + 10 \times 7 = 38 \Rightarrow a + 70 = 38 \Rightarrow a = 38 - 70 \Rightarrow a = -32 \)
We have \( a = -32 \) and \( d = 7 \). Therefore, \( a_{31} = a + (31 - 1)d = a + 30d \)
\( \Rightarrow a_{31} = (-32) + 30 \times 7 = -32 + 210 \Rightarrow a_{31} = 178 \)

Question. The first term of an AP is \( x \) and its common difference is \( y \). Find its \( 12^{th} \) term.
Answer: \( a_{12} = a + 11d = x + 11y \).

Question. In an AP, if \( d = - 4, n = 7, a_n = 4 \), then \( a \) is
(a) 6
(b) 7
(c) 20
(d) 28
Answer: (d)

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

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

Question. If \( \frac{4}{5}, a, 2 \) are three consecutive terms of an AP, then the value of \( a \) is
(a) \( \frac{5}{2} \)
(b) \( \frac{2}{7} \)
(c) \( \frac{5}{7} \)
(d) \( \frac{7}{5} \)
Answer: (d)

Question. Assertion (A): Common difference of the AP: \( -5, -1, 3, 7, \dots \) is 4.
Reason (R): Common difference of the AP : \( a, a + d, a + 2d, \dots \) is given by \( d = 2^{nd} \text{ term} - 1^{st} \text{ term} \).
(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 (A): If \( n^{th} \) term of an AP is \( 7 - 4n \), then its common difference is \( - 4 \).
Reason (R): Common difference of an AP is given by \( d = a_{n+1} - a_n \).
(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: (a)

Question. Assertion (A): Common difference of an AP in which \( a_{21} - a_7 = 84 \) is 14.
Reason (R): \( n^{th} \) term of an AP is given by \( a_n = a + (n - 1) d \).
(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. Write first four terms of the AP, whose first term and the common difference are given as follows: \( a = 10, d = 10 \)
Answer: 10, 20, 30, 40

Question. Find the \( 10^{th} \) term of the AP: 2, 7, 12, \dots
Answer: 47

Question. In the given AP, find the missing terms: \( \dots, 13, \dots, 3 \).
Answer: 18, 8

Question. Find the \( 6^{th} \) term from the end of the AP: 17, 14, 11, \dots, \( - 40 \).
Answer: \( - 25 \)

Question. Which term of the AP: 21, 18, 15, \dots is zero?
Answer: 8

Question. Write the next term of the AP: \( \sqrt{8}, \sqrt{18}, \sqrt{32}, \dots \)
Answer: \( \sqrt{50} \) or \( 5\sqrt{2} \)

Question. Find \( a, b \), and \( c \) such that the numbers \( a, 7, b, 23, c \) are in AP.
Answer: \( a = -1, b = 15, c = 31 \)