geometric-progressions MCQs for UPSC Prelims

25 practice questions on geometric-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.

7 Easy 13 Medium 5 Hard

Sample questions

Q1
hard
Assertion (A): plotting the terms of the geometric progression $16,\ 8,\ 4,\ 2,\ 1$ against their position numbers $1,\ 2,\ 3,\ 4,\ 5$ gives five points that do not lie on one straight line. Reason (R): the drops between consecutive terms are $8,\ 4,\ 2$ and $1$, which are not all equal.
  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: $16 - 8 = 8$, $8 - 4 = 4$, $4 - 2 = 2$ and $2 - 1 = 1$. A straight line needs the same change for each step of one position, so unequal drops force the points off any single line, making (A) true for exactly the reason (R) gives.
Q2
easy
A Sierpinski triangle is built like this. Stage $1$ is a single black triangle. At every later stage each black triangle is cut into four equal smaller triangles and the middle one is removed, leaving three black triangles where there was one. How many black triangles are there at Stage $3$?
  1. A $4$
  2. B $9$
  3. C $27$
  4. D $3$
Show answer and explanation

Correct answer: B - $9$

Each stage multiplies the count by $3$, so the counts run $1$, then $3$, then $9$. The value $3$ belongs to Stage $2$ and $27$ to Stage $4$, each one stage out, while $4$ counts the pieces one triangle is cut into before the middle one is thrown away.
Q3
medium
In a geometric progression the second term is $18$ and the third term is $54$. What is its first term?
  1. A $9$
  2. B $36$
  3. C $162$
  4. D $6$
Show answer and explanation

Correct answer: D - $6$

The common ratio is $54 \div 18 = 3$, so the first term is $18 \div 3 = 6$. Halving $18$ gives $9$ and treats the progression as though it doubled, $36 = 54 - 18$ uses a difference instead of a ratio, and $162$ is the fourth term found by going forwards.
Q4
medium
A geometric progression has the term in position $n$ given by $3 \times 2^{n-1}$. Which term of this progression is $384$?
  1. A $384$ is not a term of it
  2. B The $7$th term
  3. C The $9$th term
  4. D The $8$th term
Show answer and explanation

Correct answer: D - The $8$th term

Dividing gives $384 \div 3 = 128 = 2^7$, so $n - 1 = 7$ and $n = 8$. Position $7$ holds $3 \times 64 = 192$ and position $9$ holds $3 \times 256 = 768$. Since the division came out an exact power of $2$, the number certainly is a term.
Q5
easy
For the geometric progression $3,\ 6,\ 12,\ 24,\ \ldots$, what are the first term and the common ratio?
  1. A First term $3$, common ratio $3$
  2. B First term $3$, common ratio $2$
  3. C First term $3$, common ratio $6$
  4. D First term $2$, common ratio $3$
Show answer and explanation

Correct answer: B - First term $3$, common ratio $2$

The list starts at $3$, and $6 \div 3 = 2$ with $12 \div 6 = 2$, so the common ratio is $2$. Reading the ratio as $3$ repeats the first term, reading it as $6$ copies the second term, and swapping the two numbers puts the ratio in the starting place.
Q6
medium
Assertion (A): the sequence $7,\ 7,\ 7,\ 7,\ \ldots$, in which every term is $7$, is a geometric progression. Reason (R): each term of that sequence is obtained by multiplying the term before it by $0$.
  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: C - A is true, but R is false

(A) is true, because every term is the one before it multiplied by the fixed number $1$, and a common ratio of $1$ is allowed. (R) is false: multiplying $7$ by $0$ would give $0$, not $7$, so the stated multiplier is wrong even though the claim it is offered for is right.
Q7
medium
A message starts with $3$ people. In every round, each person who received it in the previous round passes it to $2$ new people, so the numbers receiving it in rounds $1,\ 2,\ 3,\ \ldots$ are $3,\ 6,\ 12,\ \ldots$, that is $3 \times 2^{n-1}$ in round $n$. How many people receive it in round $7$?
  1. A $42$
  2. B $96$
  3. C $192$
  4. D $384$
Show answer and explanation

Correct answer: C - $192$

Putting $n = 7$ gives $3 \times 2^6 = 3 \times 64 = 192$ people. The value $384$ uses $2^7$ and belongs to round $8$, $96$ uses $2^5$ and belongs to round $6$, and $42 = 3 \times 14$ treats the growth as though it added rather than doubled.
Q8
hard
A geometric progression has first term $5$, and its fourth term is $135$. What are its common ratio and its fifth term?
  1. A Ratio $27$ and fifth term $3645$
  2. B Ratio $3$ and fifth term $405$
  3. C Ratio $9$ and fifth term $1215$
  4. D Ratio $3$ and fifth term $270$
Show answer and explanation

Correct answer: B - Ratio $3$ and fifth term $405$

Three multiplications take $5$ to $135$, so the ratio cubed is $135 \div 5 = 27$, giving a ratio of $3$; the progression is $5,\ 15,\ 45,\ 135,\ 405$. Doubling $135$ gives $270$ and uses the wrong ratio at the last step, while ratios of $27$ and $9$ do not carry $5$ to $135$ in three steps.

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