Arithmetic Progressions

CBSE Class 10 — Maths — Previous Year Questions

Q1.MCQ1 mark(2020)Easy

The value of $x$ for which $2x$, $(x+10)$ and $(3x+2)$ are the three consecutive terms of an AP, is:

Q2.MCQ1 mark(2020)Easy

The first term of an AP is $p$ and the common difference is $q$, then its $10^{\text{th}}$ term is:

Q3.MCQ1 mark(2020)Easy

The common difference of the A.P. $\frac{1}{p},\frac{1-p}{p},\frac{1-2p}{p},\dots$ is

Q4.MCQ1 mark(2020)Easy

The $n^{th}$ term of the A.P. $a,3a,5a,\dots$ is

Q5.MCQ1 mark(2020)Easy

The common difference of an AP, whose $n^{th}$ term is $a_n=(3n+7)$, is

Q6.MCQ1 mark(2020)Easy

The value of $p$ for which $(2p+1)$, $10$ and $(5p+5)$ are three consecutive terms of an AP is

Q7.MCQ1 mark(2020)Easy

The number of terms of an AP $5,9,13,\dots 185$ is

Q8.MCQ1 mark(2020)Easy

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

Q9.MCQ1 mark(2020)Easy

The $9^{th}$ term of the A.P. $-15,-11,-7,\dots ,49$ is

Q10.MCQ1 mark(2020)Easy

Which of the following is *not* an A.P.?

Q11.VSA1 mark(2020)Easy

Find the sum of the first 100 natural numbers.

Q12.MCQ1 mark(2023)Easy

If $p-1$, $p+1$ and $2p+3$ are in A.P., then the value of $p$ is
(A) $-2$
(B) $4$
(C) $0$
(D) $2$

Q13.MCQ1 mark(2023)Easy

The next term of the A.P.: $\sqrt{6}$, $\sqrt{24}$, $\sqrt{54}$ is:

Q14.Assertion-Reason1 mark(2023)Medium

Assertion (A): $a$, $b$, $c$ are in A.P. if and only if $2b=a+c$.

Reason (R): The sum of first $n$ odd natural numbers is $n^2$.

Q15.MCQ1 mark(2023)Easy

The common difference of the A.P. whose $n^{th}$ term is given by $a_n=3n+7$, is:

Q16.MCQ1 mark(2023)Easy

The $11^{th}$ term from the end of the A.P.: $10,7,4,\dots ,-62$ is:

Q17.MCQ1 mark(2023)Easy

If the sum of the first $n$ terms of an A.P. be $3n^2+n$ and its common difference is $6$, then its first term is

Q18.MCQ1 mark(2024)Easy

In an A.P., if the first term $a=7$, $n$th term $a_n=84$ and the sum of first $n$ terms $s_n=\frac{2093}{2}$, then $n$ is equal to:

Q19.MCQ1 mark(2024)Easy

The common difference of an A.P. in which $a_{15}-a_{11}=48$, is

Q20.MCQ1 mark(2024)Easy

Which term of the A.P. $-29,-26,-23,\dots ,61$ is $16$?

Q21.MCQ1 mark(2024)Easy

Three numbers in A.P. have the sum $30$. What is its middle term?

Q22.MCQ1 mark(2024)Easy

The next (4th) term of the A.P. $\sqrt{18}$, $\sqrt{50}$, $\sqrt{98}$, ... is:

Q23.MCQ1 mark(2024)Easy

The number of terms in the A.P. $3,6,9,12,\dots ,111$ is:

Q24.MCQ1 mark(2025)Easy

If the sum of first $m$ terms of an AP is $2m^2+3m$, then its second term is:

Q25.Case-Study1 mark(2025)Easy

A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10.



(i) Write the fourth, fifth and sixth term of the Arithmetic Progression so formed.

Q26.Case-Study1 mark(2025)Easy

(ii) Determine the distance of the 8th round.

Q27.MCQ1 mark(2025)Easy

The $10^{th}$ term of the AP $5,\frac{19}{4},\frac{9}{2},\frac{17}{4},\dots$ is:

Q28.Assertion-Reason1 mark(2025)Easy

Assertion (A): Common difference of the AP $5,1,-3,-7,\dots$ is $4$.

Reason (R): Common difference of the AP $a_1,a_2,a_3,\dots ,a_n$ is obtained by $d=a_n-a_{n-1}$.

Q29.VSA2 marks(2020)Medium

Show that $(a-b{)}^2$, $(a^2+b^2)$ and $(a+b{)}^2$ are in AP.

Q30.VSA2 marks(2020)Easy

Find the $11^{th}$ term from the last term (towards the first term) of the AP $12,8,4,\dots ,-84$.

Q31.VSA2 marks(2020)Medium

Solve the equation: $1+5+9+13+\dots +x=1326$.

Q32.VSA2 marks(2022)Easy

Find the sum of first 30 terms of AP: $-30,-24,-18,...$

Q33.VSA2 marks(2022)Medium

In an AP if $S_n=n(4n+1)$, then find the AP.

Q34.VSA2 marks(2022)Easy

Which term of the A.P. $\frac{-11}{2},-3,\frac{-1}{2},\dots$ is $\frac{49}{2}$?

Q35.VSA2 marks(2022)Easy

Find $a$ and $b$ so that the numbers $a,7,b,23$ are in A.P.

Q36.VSA2 marks(2022)Easy

Find the sum of first $20$ terms of an A.P. whose $n^{th}$ term is given as $a_n=5-2n$.

Q37.VSA2 marks(2022)Medium

How many natural numbers are there between 1 and 1000 which are divisible by 5 but not by 2?

Q38.Case-Study2 marks(2025)Easy

(iii) (a) Find the total distance run after completing all 10 rounds.

Q39.Case-Study2 marks(2025)Easy

(iii) (b) If a runner completes only the first 6 rounds, what is the total distance run by the runner?

Q40.SA3 marks(2020)Medium

Show that the sum of all terms of an A.P. whose first term is $a$, the second term is $b$ and the last term is $c$ is equal to $\frac{(a+c)(b+c-2a)}{2(b-a)}$.

Q41.SA3 marks(2020)Medium

Solve the equation: $1+4+7+10+\dots +x=287$.

Q42.SA3 marks(2020)Easy

Find $a$, $b$ and $c$ if it is given that the numbers $a,7,b,23,c$ are in AP.

Q43.SA3 marks(2020)Medium

If $m$ times the $m^{th}$ term of an AP is equal to $n$ times its $n^{th}$ term, show that the $(m+n{)}^{th}$ term of the AP is zero.

Q44.SA3 marks(2020)Medium

The sum of the first 30 terms of an A.P. is 1920. If the fourth term is 18, find its $11^{th}$ term.

Q45.SA3 marks(2020)Medium

For an A.P., it is given that the first term $(a)=5$, common difference $(d)=3$, and the $n^{th}$ term $(a_n)=50$. Find $n$ and sum of first $n$ terms $(S_n)$ of the A.P.

Q46.SA3 marks(2022)Medium

If the last term of an A.P. of 30 terms is 119 and the $8^{th}$ term from the end (towards the first term) is 91, then find the common difference of the A.P. Hence, find the sum of all the terms of the A.P.

Q47.SA3 marks(2023)Easy

How many terms are there in an A.P. whose first and fifth terms are $-14$ and $2$, respectively and the last term is $62$.

Q48.SA3 marks(2023)Medium

Which term of the A.P.: $65,61,57,53,\dots$ is the first negative term?

Q49.SA3 marks(2023)Medium

If $p^{th}$ term of an A.P. is $q$ and $q^{th}$ term is $p$, then prove that its $n^{th}$ term is $(p+q-n)$.

Q50.SA3 marks(2024)Hard

If the sum of first $m$ terms of an A.P. is same as sum of its first $n$ terms $(m\ne n)$, then show that the sum of its first $(m+n)$ terms is zero.

Q51.SA3 marks(2024)Medium

In an A.P., the sum of three consecutive terms is $24$ and the sum of their squares is $194$. Find the numbers.

Q52.SA3 marks(2024)Medium

If the sum of the first 14 terms of an A.P. is 1050 and the first term is 10, then find the 20th term and the $n^{\text{th}}$ term.

Q53.SA3 marks(2024)Medium

The first term of an A.P. is 5, the last term is 45 and the sum of all the terms is 400. Find the number of terms and the common difference of the A.P.

Q54.SA3 marks(2025)Medium

A sum of $\text{rupee}2000$ is invested at $7\%$ per annum simple interest. Calculate the interest at the end of $1^{st}$, $2^{nd}$ and $3^{rd}$ year. Do these interests form an AP? If so, find the interest at the end of the $27^{th}$ year.

Q55.LA4 marks(2020)Hard

The sum of four consecutive numbers in an AP is $32$ and the ratio of product of the first and last terms to the product of two middle terms is $7:15$. Find the numbers.

Q56.LA4 marks(2020)Medium

Solve: $1+4+7+10+\dots +x=287$

Q57.Case-Study4 marks(2022)Medium

In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in different figures.

One such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression:




(a) Write the AP for the number of triangles used in the figures. Also, write the $n^{th}$ term of this AP.

(b) Which figure has $61$ matchsticks?

Q58.Case-Study4 marks(2024)Medium

Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize.

While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{\text{th}}$ spot is $20+4n$, then answer the following questions to help the players in spotting the clues:



(i) Which number is on first spot?
(ii) (a) Which spot is numbered as $112$?

**OR**

(ii) (b) What is the sum of all the numbers on the first $10$ spots?
(iii) Which number is on the $(n-2{)}^{\text{th}}$ spot?

Q59.Case-Study4 marks(2024)Medium

Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize.

While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{\text{th}}$ spot is $20+4n$, then answer the following questions to help the players in spotting the clues:



(i) Which number is on first spot?
(ii) (b) What is the sum of all the numbers on the first $10$ spots?
(iii) Which number is on the $(n-2{)}^{\text{th}}$ spot?

Q60.Case-Study4 marks(2025)Medium

Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of cable car ride from base point to top most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which car moves.

The distance of first pole from base point is 200 m and subsequent poles are installed at equal interval of 150 m. Further, the distance of last pole from the top is 300 m.



Based on above information, answer the following questions using Arithmetic Progression:
(i) Find the distance of $10^{\text{th}}$ pole from the base.
(ii) Find the distance between $15^{\text{th}}$ pole and $25^{\text{th}}$ pole.
(iii) (a) Find the time taken by cable car to reach $15^{\text{th}}$ pole from the top if it is moving at the speed of 5 m/sec and coming from top.

OR

(iii) (b) Find the total number of poles installed along the entire journey.

Q61.LA5 marks(2023)Hard

The ratio of the $11^{th}$ term to the $17^{th}$ term of an A.P. is $3:4$. Find the ratio of the $5^{th}$ term to the $21^{st}$ term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.

Q62.LA5 marks(2023)Medium

250 logs are stacked in the following manner: 22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows, are the 250 logs placed and how many logs are there in the top row?

Q63.LA5 marks(2023)Medium

How many terms of the arithmetic progression $45,39,33,\dots$ must be taken so that their sum is 180? Explain the double answer.

Q64.LA5 marks(2023)Hard

The ratio of the $11^{th}$ term to the $18^{th}$ term of an A.P. is $2:3$. Find the ratio of the $5^{th}$ term to the $21^{st}$ term, and also the ratio of the sum of first $5$ terms to the sum of first $21$ terms.

Q65.LA5 marks(2023)Medium

If the sum of first $6$ terms of an A.P. is $36$ and that of the first $16$ terms is $256$, find the sum of first $10$ terms.

Q66.LA5 marks(2024)Hard

The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms.

Q67.LA5 marks(2024)Hard

In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the first term and common difference of the A.P. Also, find the sum of all the terms of the A.P.

Q68.LA5 marks(2025)Hard

The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.

Q69.LA5 marks(2025)Hard

The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.