arithmetic-progressions MCQs for UPSC Prelims

25 practice questions on arithmetic-progressions from the Predicting What Comes Next: Exploring Sequences and Progressions section of the UPSC Prelims syllabus. 25 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

8 Easy 12 Medium 5 Hard

Sample questions

Q1
easy
An arithmetic progression has first term $6$ and common difference $4$. What is its $10$th term?
  1. A $46$
  2. B $38$
  3. C $40$
  4. D $42$
Show answer and explanation

Correct answer: D - $42$

Nine steps of $4$ from the first term give $6 + 9 \times 4 = 6 + 36 = 42$. The value $46$ takes ten steps instead of nine, $38$ takes only eight, and $40 = 10 \times 4$ leaves out the first term completely.
Q2
medium
Assertion (A): the sequence of square numbers $1,\ 4,\ 9,\ 16,\ 25,\ \ldots$ is an arithmetic progression. Reason (R): the differences between its consecutive terms are $3,\ 5,\ 7,\ 9,\ \ldots$.
  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: D - A is false, but R is true

(R) is correct: $4-1=3$, $9-4=5$, $16-9=7$ and $25-16=9$. Those differences keep growing rather than staying fixed, so the square numbers do not form an arithmetic progression and (A) is false. The true reason (R) is the very evidence that sinks (A).
Q3
medium
Assertion (A): the sequence $5,\ 5,\ 5,\ 5,\ \ldots$, in which every term is $5$, is an arithmetic progression. Reason (R): in that sequence the difference between each term and the one before it is $0$, the same all the way along.
  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: A - Both A and R are true, and R is the correct explanation of A

(R) is correct, since $5 - 5 = 0$ at every step. An arithmetic progression only needs its consecutive differences to be equal, and a common difference of $0$ satisfies that, so (A) is true as well and (R) is exactly the reason why.
Q4
medium
The first term of an arithmetic progression is $3$ and it climbs by $7$ each time. Which of its terms is $80$?
  1. A The $12$th term
  2. B The $11$th term
  3. C The $13$th term
  4. D $80$ is not a term of it
Show answer and explanation

Correct answer: A - The $12$th term

Solving $3 + (n-1) \times 7 = 80$ gives $(n-1) \times 7 = 77$, so $n - 1 = 11$ and $n = 12$. Position $11$ holds $73$ and position $13$ holds $87$. Since $n$ came out a whole number, $80$ certainly is a term.
Q5
easy
For the arithmetic progression $9,\ 15,\ 21,\ 27,\ \ldots$, what are the first term $a$ and the common difference $d$?
  1. A $a = 6$ and $d = 9$
  2. B $a = 9$ and $d = 6$
  3. C $a = 9$ and $d = 15$
  4. D $a = 15$ and $d = 6$
Show answer and explanation

Correct answer: B - $a = 9$ and $d = 6$

The list starts at $9$, so $a = 9$, and $15 - 9 = 6$ with $21 - 15 = 6$, so $d = 6$. Taking $d = 15$ copies the second term instead of the gap, while the other two swap the roles of the first term and the common difference or start the count from the second term.
Q6
hard
A growing pattern of matchsticks uses $4$ matchsticks at Stage $1$, $7$ at Stage $2$ and $10$ at Stage $3$, with each stage using $3$ more than the stage before. At which stage does the pattern first use more than $100$ matchsticks?
  1. A Stage $32$
  2. B Stage $35$
  3. C Stage $34$
  4. D Stage $33$
Show answer and explanation

Correct answer: C - Stage $34$

The count at Stage $n$ is $3n + 1$. Stage $33$ uses exactly $100$, which is not more than $100$, and Stage $34$ uses $103$, so Stage $34$ is the first to pass it. Stage $32$ uses $97$, and Stage $35$ passes $100$ but is not the first stage to do so.
Q7
easy
Three of these four sequences are arithmetic progressions. Which one is not?
  1. A $3,\ 8,\ 13,\ 18,\ 23$
  2. B $20,\ 17,\ 14,\ 11,\ 8$
  3. C $-6,\ -2,\ 2,\ 6,\ 10$
  4. D $1,\ 2,\ 4,\ 8,\ 16$
Show answer and explanation

Correct answer: D - $1,\ 2,\ 4,\ 8,\ 16$

In $1,\ 2,\ 4,\ 8,\ 16$ the gaps are $1,\ 2,\ 4,\ 8$, which keep changing, so it is not an arithmetic progression. The other three have constant gaps of $5$, $-3$ and $4$, and a negative gap or a negative starting value is perfectly allowed.
Q8
hard
In an arithmetic progression the fifth term is $28$ and the eighth term is $43$. What are its first term and its common difference?
  1. A $a = 8$ and $d = 5$
  2. B $a = 3$ and $d = 5$
  3. C $a = 8$ and $d = 3$
  4. D $a = 13$ and $d = 5$
Show answer and explanation

Correct answer: A - $a = 8$ and $d = 5$

Three steps carry $28$ up to $43$, a rise of $15$, so $d = 15 \div 3 = 5$. Stepping back four times from the fifth term gives $28 - 4 \times 5 = 8$, so $a = 8$. Taking $a = 13$ steps back only three times, $d = 3$ comes from dividing by $5$ instead of $3$, and $a = 3$ steps back five times.

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