CBSE Class 10 Arithmetic Progressions Sure Shot Questions Set 01

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Q1. Determine the AP whose 3^rd term is 5 and the 7^th term is 9.

Q2. The 8^th term of an AP is 37 and its 12^th term is 57. Find the AP.

Q3. The 7^th term of an AP is – 4 and its 13^th term is – 16. Find the AP.

Q4. If the 10^th term of an AP is 52 and the 17^th term is 20 more than the 13^th term, find the AP.

Q5. If the 8^th term of an AP is 31 and its 15^th term is 16 more than the 11^th term, find the AP.

Q6. Check whether 51 is a term of the AP 5, 8, 11, 14, ……?

Q7. The 6th term of an AP is – 10 and its 10^th term is – 26. Determine the 15^th term of the AP.

Q8. The sum of 4^th term and 8th term of an AP is 24 and the sum of 6^th and 10^th terms is 44. Find the AP.

Q9. The sum of 5^th term and 9^th term of an AP is 72 and the sum of 7^th and 12^th terms is 97. Find the AP.

Q14. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29^th term.

Q15. Determine the AP whose third term is 16 and the 7^th term exceeds the 5th term by 12.

Q16. The 17^th term of an AP exceeds its 10^th term by 7. Find the common difference.

Q17. If the nth term of an AP is (5n – 2), find its first term and common difference. Also find its 19^th term.

Q18. If the nth term of an AP is (4n – 10), find its first term and common difference. Also find its 16^th term.

Q19. If 2x, x + 10, 3x + 2 are in A.P., find the value of x.

Recap Notes

SEQUENCE

• Sequence is a list of numbers that follows a certain rule. Each of the numbers in the list is called a term. For example, 1, 9, 13, 18, 27, ... , it is a sequence where numbers are arranged in ascending order.

PROGRESSION

• The sequence, whose terms always follow a certain pattern is called progression.

ARITHMETIC PROGRESSION (A.P.)

• A progression is called an arithmetic progression if the difference between its any two consecutive terms is a fixed constant called common difference denoted by \(d\).
• i.e., Let \(a_1, a_2, a_3, a_4, ......., a_n\) be an arithmetic progression. Then, \(a_2 – a_1= a_3 – a_2 = a_4 – a_3 = .......... = a_n – a_{n – 1} = d\)
• Common difference can be negative, zero or positive.
• \(a, a + d, a + 2d, a + 3d, ...\) represents an arithmetic progression where \(a\) is the first term and \(d\) is the common difference. This is known as general form of an A.P.
• In general, for an A.P. \(a_1, a_2, a_3, ..., a_n\); \(d = a_{r + 1} – a_r\), where \(a_r\) and \(a_{r + 1}\) are the \(r^{th}\) and \((r + 1)^{th}\) term of the A.P. respectively. To find common difference (\(d\)) in a given A.P., we need not find all of \(a_2 – a_1, a_3 – a_2, ...\) we can find only one of them.

Note : (i) Any three numbers in A.P. can be taken as \(a – d, a, a + d\).
(ii) Any four numbers in A.P. can be taken as \(a – 3d, a – d, a + d, a + 3d\).

Types of A.P.

There are two types of A.P. as given below :

• (i) Finite A.P. : An arithmetic progression whose number of terms is finite is called finite A.P. For example, 900, 800, 700, ...., 100, it is a finite A.P.
• (ii) Infinite A.P. : An arithmetic progression whose number of terms is infinite (not countable) is called an infinite A.P. For example, 7, 10, 13, 16, ...., it is an infinite A.P.

GENERAL TERM OR \(n^{th}\) TERM OF AN A.P.

• Let \(a, a + d, a + 2d, .........\) represents an A.P. with first term \(a\) and common difference \(d\). The general term is denoted by \(a_n = a + (n – 1) d\), where \(n\) is a natural number.
• If the A.P. is finite i.e., if there are \(n\) terms in an A.P., then \(n^{th}\) term is known as last term of the A.P. and it is denoted by \(l\), so \(l = a + (n – 1)d\), where \(a\) is the first term and \(d\) is the common difference.

For example, \(15^{th}\) term of the A.P. 13, 9, 5, 1, –3, .... is \(a_{15} = 13 + (15 – 1)(–4)\), where \(a = 13\) and \(d = –4\) so, \(a_{15} = 13 – 4 \times 14 = –43\)

Note :
(i) \(n^{th}\) term from the end of the A. P. is \(a + (m – n) d\), where \(m\) is the total number of terms in the A.P.
(ii) \(n^{th}\) term from the end of A.P. is \(l – (n – 1)d\), where \(l\) is last term, \(d\) is common difference and \(n\) is number of terms.

SUM OF FIRST \(n\) TERMS OF AN A.P.

• For an A. P., \(a, a + d, a + 2d, ......\) sum of first \(n\) terms is denoted by \(S_n\) and defined by \(S_n = \frac{n}{2}[2a + (n – 1)d]\), where \(a\) is the first term and \(d\) is the common difference.
• It can also be written as \(S_n = \frac{n}{2}[a + l]\), where \(l\) is the last term given by \(l = a + (n – 1) d\).

For example, sum of first 20 terms of the A.P. : 3, 7, 11, 15, … is \(S_{20} = \frac{20}{2}(6 + (20 – 1) 4)\), where first term is 3 and common difference is 4.
\(\therefore S_{20} = 10(6 + 19 \times 4) = 10(82) = 820\)

Note : If \(S_n\) is the sum of first \(n\) terms of an A.P., then its \(n^{th}\) term, \(a_n\) is defined as \(a_n = S_n – S_{n – 1}\).

Multiple Choice Questions 

Question. The common difference of the A.P. \(\frac{1}{3q}, \frac{1 - 6q}{3q}, \frac{1 - 12q}{3q}, ...... \) is
(a) \(q\)
(b) \(-q\)
(c) \(-2\)
(d) 2
Answer: (c)

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

Question. The next term of the A.P. \(\sqrt{18}, \sqrt{50}, \sqrt{98} ...\) is
(a) \(\sqrt{146}\)
(b) \(\sqrt{128}\)
(c) \(\sqrt{162}\)
(d) \(\sqrt{200}\)
Answer: (c)

Question. The value of \(a_{30} – a_{20}\) for the A.P. 2, 7, 12, 17, ... is
(a) 100
(b) 10
(c) 50
(d) 20
Answer: (c)

Question. In an A.P., if \(a = -10, n = 6\) and \(a_n = 10\), then the value of \(d\) is
(a) 0
(b) 4
(c) \(-4\)
(d) 10/3
Answer: (b)

Question. If the sum of first \(m\) terms of an A.P. is \(2m^2 + 3m\), then what is its second term?
(a) 9
(b) 10
(c) 11
(d) 12
Answer: (a)

Question. If the sum of \(n\) terms of two A.P.s are in the ratio \((3n + 5) : (5n + 7)\), then their \(n^{th}\) terms are in the ratio
(a) \((3n – 1) : (5n – 1)\)
(b) \((3n + 1) : (5n – 1)\)
(c) \((3n + 1) : (5n + 1)\)
(d) \((5n + 1) : (3n + 1)\)
Answer: (a)

Question. If the \(10^{th}\) term of an A.P. is 52 and \(17^{th}\) term is 20 more than the \(13^{th}\) term, then find the A.P.
(a) 40, 45, 50,.....
(b) 45, 50, 55,.....
(c) 17, 22, 27,.....
(d) 7, 12, 17,.....
Answer: (d)

Question. Two persons Anil and Happy joined D.W. Associates. Anil and Happy started with an initial salary of ₹ 50000 and ₹ 64000 respectively with annual increment of ₹ 2500 and ₹ 2000 each respectively. In which year will Anil start earning more salary than Happy?
(a) \(28^{th}\)
(b) \(29^{th}\)
(c) \(30^{th}\)
(d) \(27^{th}\)
Answer: (b)

Question. The production of TV in a factory increases uniformly by a fixed number every year. It produced 8000 TV’s in \(6^{th}\) year & 11300 in \(9^{th}\) year, find the production in \(8^{th}\) year.
(a) 10500
(b) 9800
(c) 9700
(d) 10200
Answer: (d)

Question. The number of terms in the A.P. 3, 6, 9, 12,...., 111 is
(a) 25
(b) 40
(c) 37
(d) 30
Answer: (c)

Question. A man starts repaying a loan with first monthly installment of ₹ 1000. If he increases the installment by ₹ 50 every month, what amount will he pay in the \(30^{th}\) installment?
(a) ₹ 1450
(b) ₹ 2450
(c) ₹ 2050
(d) ₹ 2040
Answer: (b)

Question. The value of \(x\) for which \((8x + 4), (6x – 2)\) and \((2x + 7)\) are in A.P., is
(a) 15/2
(b) 2/15
(c) –15/2
(d) –2/15
Answer: (a)

Question. The numbers –11, – 7, – 3, 1, 5, ...... are
(a) in A.P. with \(d = -18\)
(b) in A.P. with \(d = -4\)
(c) in A.P. with \(d = 4\)
(d) not in A.P.
Answer: (c)

Question. Which term of the A.P. 3, 15, 27, 39, ...... will be 252 more than its \(44^{th}\) term?
(a) \(66^{th}\)
(b) \(64^{th}\)
(c) \(65^{th}\)
(d) \(67^{th}\)
Answer: (c)

Question. If \(p^{th}\) term of an A.P. is \(\frac{3p - 1}{6}\), then sum of first \(n\) terms of the A.P. is
(a) \(\frac{n}{12} [3n + 1]\)
(b) \(\frac{n}{12} [3n - 1]\)
(c) \(\frac{n}{6} [3n + 1]\)
(d) \(\frac{n}{6} [3n - 1]\)
Answer: (a)

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

Question. For what value of \(n\), are the \(n^{th}\) terms of two A.P.’s 52, 54, 56, ..... and 4, 12, 20, ..... equal ?
(a) 11
(b) 12
(c) 10
(d) 9
Answer: (d)

Question. Find the sum of all two digit natural numbers which when divided by 3 yield 1 as remainder.
(a) 1605
(b) 1780
(c) 1080
(d) 1960
Answer: (a)

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)

Question. If the common difference of an A.P. is 5, then what is \(a_{18} – a_{13}\) ?
(a) 5
(b) 20
(c) 25
(d) 30
Answer: (c)

Question. What is the common difference of four terms in an A.P. such that the ratio of the product of the first and fourth terms to that of the second and third is 2 : 3 and the sum of all four terms is 20?
(a) 3
(b) 1
(c) 4
(d) 2
Answer: (d)

Question. If the seventh term of an A.P. is 1/9 and its ninth term is 1/7, find common difference.
(a) 1
(b) 2/63
(c) 3/64
(d) 1/63
Answer: (d)

Question. The sum \((– 6) + (0) + (6) + ..... \) upto \(13^{th}\) term =
(a) 390
(b) 1380
(c) 378
(d) 1830
Answer: (a)

Question. If \(a, (a – 2)\) and \(3a\) are in A.P., then the value of \(a\) is
(a) – 3
(b) – 2
(c) 3
(d) 2
Answer: (a)

Question. If \(m^{th}\) term of an A.P. is \(1/n\) and \(n^{th}\) term is \(1/m\), then the sum of first \(mn\) terms is
(a) \(mn + 1\)
(b) \(\frac{mn + 1}{2}\)
(c) \(\frac{mn - 1}{2}\)
(d) \(\frac{mn - 1}{3}\)
Answer: (b)

Question. If 9 times the \(9^{th}\) term in an arithmetic progression is equal to 15 times of its \(15^{th}\) term, then what is the \(24^{th}\) term?
(a) 0
(b) 9
(c) 15
(d) 23
Answer: (a)

Question. If \(x \neq y\) and the sequences \(x, a_1, a_2, y\) and \(x, b_1, b_2, y\) each are in A.P., then \(\frac{a_2 - a_1}{b_2 - b_1}\) is
(a) 2/3
(b) 3/2
(c) 1
(d) 3/4
Answer: (c)

Question. If the sum of 7 terms of an A.P. is 49 and that of 17 terms is 289, then, its first term is
(a) 1
(b) – 1
(c) 2
(d) –2
Answer: (a)

Question. Find how many terms are there in the A.P. 16, 24, 32, ......, 96.
(a) 10
(b) 11
(c) 12
(d) 14
Answer: (b)

Question. If the first, second and last terms of an A.P. are \(a, b\) and \(2a\) respectively, its sum is
(a) \(\frac{ab}{2(b - a)}\)
(b) \(\frac{ab}{b - a}\)
(c) \(\frac{3ab}{2(b - a)}\)
(d) None of these
Answer: (c)

Question. Find the sum of first 15 multiples of 8.
(a) 840
(b) 1020
(c) 960
(d) 920
Answer: (c)

Question. Find the sum of first 10 terms of the A.P. \(x – 8, x – 2, x + 4, …\)
(a) \(10x + 210\)
(b) \(10x + 190\)
(c) \(5x + 190\)
(d) \(5x + 210\)
Answer: (b)

Question. In an A.P., the sum of first \(n\) terms is \(\frac{3}{2}n^2 + \frac{13}{2}n\). Find its \(15^{th}\) term.
(a) 45
(b) 50
(c) 60
(d) 75
Answer: (b)

Question. Three numbers in an A.P. have sum 18. Its middle term is
(a) 6
(b) 8
(c) 3
(d) 2
Answer: (a)

Question. Find the sixteenth term of the A.P. –10, –6, –2, 2,...
(a) 10
(b) 20
(c) 40
(d) 50
Answer: (d)

Question. \(\frac{3}{\sqrt{5}} + \sqrt{5} + \frac{7}{\sqrt{5}} + ...\) to 15 terms is equal to
(a) \(51\sqrt{5}\)
(b) \(17\sqrt{5}\)
(c) \(81\sqrt{5}\)
(d) \(9\sqrt{5}\)
Answer: (a)

Question. Which of the following is not an A.P. ?
(a) \(-3, \frac{-5}{2}, -2, \frac{-3}{2},......\)
(b) 0.3, 0.33, 0.333, ......
(c) \(\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, .....\)
(d) \(p, 2p + 1, 3p + 2, 4p + 3, .....\)
Answer: (b)