Introduction
Arithmetic Progressions (A.P.) form one of the most important topics in Class 10 Mathematics. To help students practice effectively, here is a well-organized list of extra questions collected from advanced reference books and exemplar problems. These questions will strengthen conceptual understanding and improve problem-solving skills.
Questions
- Find the sum of all 11 terms of an A.P. whose middle term is 30.
- If the m-th term of an A.P. is 1/n and the n-th term is 1/m, show that the sum of mn terms is (mn/2)(m + n + 1).
- The sums of n, 2n and 3n terms of an A.P. are S1, S2 and S3 respectively. Prove that S3 = 3(S2 – S1).
- Show that the sum of an A.P. whose first term is a, second term is b and last term is c is equal to [(a + c)(b + c – 2a)] / [2(b – a)].
- Solve the equation: 1 + 4 + 7 + 10 + … + x = 287.
- A construction job has a penalty where the fine for the first day is ₹200, for the second day ₹250, for the third day ₹300, and so on. If the penalty increases by ₹50 every day, find the penalty for 30 days of delay.
- The digits of a 3-digit number are in A.P. and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.
- A man repays a loan of ₹3250 by paying ₹20 in the first month and increasing each payment by ₹15 every month. How long will it take him to clear the loan?
- The ratio of the m-th terms of two A.P.s is (14m – 6) : (8m + 23). Find the ratio of their n-th terms.
- The ratio of the sums of m and n terms of an A.P. is m² : n². Show that the ratio of their m-th and n-th terms is (2m – 1) : (2n – 1).
- If an A.P. has (2n + 1) terms, show that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n.
- The sums of n terms of three A.P.s are S1, S2 and S3. The first term of each is 1, and their common differences are 1, 2 and 3 respectively. Prove that S1 + S3 = 2S2.
- The sum of the 3rd and 7th terms of an A.P. is 6 and their product is 8. Find the sum of the first 16 terms.
- If the sum of m terms of an A.P. is n and the sum of n terms is m, prove that the sum of (m + n) terms is –(m + n).
- If the sum of m terms is equal to the sum of n terms of an A.P., prove that the sum of (m + n) terms is zero.
- The sum of the first p, q and r terms of an A.P. are a, b and c respectively. Show that (a/p)(q – r) + (b/q)(r – p) + (c/r)(p – q) = 0.
- The ratio of the sum of n terms of two A.P.s is (7n + 1) : (4n + 27). Find the ratio of their n-th terms.
- Find the sum of all 3-digit natural numbers that are divisible by 7.
- Find the sum of all natural numbers between 250 and 1000 which are divisible by 3.
- Find the sum of all odd integers between 2 and 100 that are divisible by 3.
- A sum of ₹280 is to be used for awarding four prizes. Each prize is ₹20 less than the previous one. Find the value of each prize.
- In a school, students decided to plant trees. Class 1 plants 1 tree, Class 2 plants 2 trees, …, Class 12 plants 12 trees. Each class has 3 sections. How many trees were planted in total?
- An A.P. consists of 37 terms. The sum of the three middle terms is 225 and the sum of the last three terms is 429. Find the A.P.
- If 2x, x + 10 and 3x + 2 are in A.P., find the value of x.
- The sum of three numbers in A.P. is –3 and their product is 8. Find the numbers.
- Find four numbers in A.P. whose sum is 20 and the sum of their squares is 120.
- Divide 32 into four parts in A.P. such that the product of the extremes and the product of the means are in the ratio 7 : 15.