Class 10 › Mathematics Arithmetic Progressions Class 10 Mathematics · Chapter 5
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View all 15 amazing Marks Booster kits → An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number, the common difference, to the previous term. The chapter derives two central formulae: the nth term \(a_n = a + (n-1)d\), and the sum of the first \(n\) terms \(S_n = \dfrac{n}{2}\,[2a + (n-1)d]\) (equivalently \(\dfrac{n}{2}(a + l)\) when the last term \(l\) is known). These let students find any term, count terms, and add long sequences quickly. APs model many real situations such as savings, salaries and seating arrangements.
Arithmetic progression
A sequence with a constant difference between consecutive terms.
Common difference (d)
The fixed amount added to get the next term.
nth term
\(a_n = a + (n - 1)d\).
Sum of n terms
\(S_n = \dfrac{n}{2}\,[2a + (n - 1)d]\).
Q1 Find the 10th term of the AP: 2, 5, 8, … Answer Q2 Write the formula for the sum of the first \(n\) terms of an AP. Answer Q3 What is an arithmetic progression (AP)? Answer Q4 Write the formula for the nth term of an AP. Answer Q5 Write the formula for the sum of the first n terms of an AP. Answer Q6 Find the common difference and the 10th term of the AP 3, 7, 11, … Answer Q7 How many two-digit numbers are divisible by 3? Answer Q8 Find the sum of the first 20 terms of the AP 2, 5, 8, … Answer Q9 Which term of the AP 5, 9, 13, … is 81? Answer Q10 Is 0 a term of the AP 31, 28, 25, …? Explain. Answer Q1 1. (i) The taxi fare after each km when the fare is \(\text{₹}\,15\) for the first km and \(\text{₹}\,8\) for each additional km. Answer Q2 1. (ii) The amount of air present in a cylinder when a vacuum pump removes \(\frac{1}{4}\) of the air remaining in the cylinder at a time. Answer Q3 1. (iii) The cost of digging a well after every metre of digging, when it costs \(\text{₹}\,150\) for the first metre and rises by \(\text{₹}\,50\) for each subsequent metre. Answer Q4 1. (iv) The amount of money in the account every year, when \(\text{₹}\,10000\) is deposited at compound interest at \(8\%\) per annum. Answer Q5 2. (i) \(a=10,\ d=10\) Answer Q6 2. (ii) \(a=-2,\ d=0\) Answer Q7 2. (iii) \(a=4,\ d=-3\) Answer Q8 2. (iv) \(a=-1,\ d=\frac{1}{2}\) Answer Q9 2. (v) \(a=-1.25,\ d=-0.25\) Answer Q10 3. (i) \(3,1,-1,-3,\ldots\) Answer Q11 3. (ii) \(-5,-1,3,7,\ldots\) Answer Q12 3. (iii) \(\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},\ldots\) Answer Q13 3. (iv) \(0.6,1.7,2.8,3.9,\ldots\) Answer Q14 4. (i) \(2,4,8,16,\ldots\) Answer Q15 4. (ii) \(2,\frac{5}{2},3,\frac{7}{2},\ldots\) Answer Q16 4. (iii) \(-1.2,-3.2,-5.2,-7.2,\ldots\) Answer Q17 4. (iv) \(-10,-6,-2,2,\ldots\) Answer Q18 4. (v) \(3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},\ldots\) Answer Q19 4. (vi) \(0.2,0.22,0.222,0.2222,\ldots\) Answer Q20 4. (vii) \(0,-4,-8,-12,\ldots\) Answer Q21 4. (viii) \(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\ldots\) Answer Q22 4. (ix) \(1,3,9,27,\ldots\) Answer Q23 4. (x) \(a,2a,3a,4a,\ldots\) Answer Q24 4. (xi) \(a,a^2,a^3,a^4,\ldots\) Answer Q25 4. (xii) \(\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},\ldots\) Answer Q26 4. (xiii) \(\sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12},\ldots\) Answer Q27 4. (xiv) \(1^2,3^2,5^2,7^2,\ldots\) Answer Q28 4. (xv) \(1^2,5^2,7^2,73,\ldots\) Answer Q1 1. (i) \(a=7,\ d=3,\ n=8,\ a_n=\ldots\) Answer Q2 1. (ii) \(a=-18,\ d=\ldots,\ n=10,\ a_n=0\) Answer Q3 1. (iii) \(a=\ldots,\ d=-3,\ n=18,\ a_n=-5\) Answer Q4 1. (iv) \(a=-18.9,\ d=2.5,\ n=\ldots,\ a_n=3.6\) Answer Q5 1. (v) \(a=3.5,\ d=0,\ n=105,\ a_n=\ldots\) Answer Q6 2. (i) \(30\)th term of the AP: \(10,7,4,\ldots\), is (A) \(97\) (B) \(77\) (C) \(-77\) (D) \(-87\) Answer Q7 2. (ii) \(11\)th term of the AP: \(-3,-\frac{1}{2},2,\ldots\), is (A) \(28\) (B) \(22\) (C) \(-38\) (D) \(-48\frac{1}{2}\) Answer Q8 3. (i) \(2,\square,26\) Answer Q9 3. (ii) \(\square,13,\square,3\) Answer Q10 3. (iii) \(5,\square,\square,9\frac{1}{2}\) Answer Q11 3. (iv) \(-4,\square,\square,\square,\square,6\) Answer Q12 3. (v) \(\square,38,\square,\square,\square,-22\) Answer Q13 4. Which term of the AP : \(3,8,13,18,\ldots\), is \(78\)? Answer Q14 5. (i) \(7,13,19,\ldots,205\) Answer Q15 5. (ii) \(18,15\frac{1}{2},13,\ldots,-47\) Answer Q16 6. Check whether \(-150\) is a term of the AP : \(11,8,5,2,\ldots\) Answer Q17 7. Find the \(31\)st term of an AP whose \(11\)th term is \(38\) and the \(16\)th term is \(73\). Answer Q18 8. An AP consists of \(50\) terms of which \(3\)rd term is \(12\) and the last term is \(106\). Find the \(29\)th term. Answer Q19 9. If the \(3\)rd and the \(9\)th terms of an AP are \(4\) and \(-8\) respectively, which term of this AP is zero? Answer Q20 10. The \(17\)th term of an AP exceeds its \(10\)th term by \(7\). Find the common difference. Answer Q21 11. Which term of the AP : \(3,15,27,39,\ldots\) will be \(132\) more than its \(54\)th term? Answer Q22 12. Two APs have the same common difference. The difference between their \(100\)th terms is \(100\), what is the difference between their \(1000\)th terms? Answer Q23 13. How many three-digit numbers are divisible by \(7\)? Answer Q24 14. How many multiples of \(4\) lie between \(10\) and \(250\)? Answer Q25 15. For what value of \(n\), are the \(n\)th terms of two APs: \(63,65,67,\ldots\) and \(3,10,17,\ldots\) equal? Answer Q26 16. Determine the AP whose third term is \(16\) and the \(7\)th term exceeds the \(5\)th term by \(12\). Answer Q27 17. Find the \(20\)th term from the last term of the AP : \(3,8,13,\ldots,253\). Answer Q28 18. The sum of the \(4\)th and \(8\)th terms of an AP is \(24\) and the sum of the \(6\)th and \(10\)th terms is \(44\). Find the first three terms of the AP. Answer Q29 19. Subba Rao started work in \(1995\) at an annual salary of \(\text{₹}\,5000\) and received an increment of \(\text{₹}\,200\) each year. In which year did his income reach \(\text{₹}\,7000\)? Answer Q30 20. Ramkali saved \(\text{₹}\,5\) in the first week of a year and then increased her weekly savings by \(\text{₹}\,1.75\). If in the \(n\)th week, her weekly savings become \(\text{₹}\,20.75\), find \(n\). Answer Q1 1. (i) \(2,7,12,\ldots\), to \(10\) terms. Answer Q2 1. (ii) \(-37,-33,-29,\ldots\), to \(12\) terms. Answer Q3 1. (iii) \(0.6,1.7,2.8,\ldots\), to \(100\) terms. Answer Q4 1. (iv) \(\frac{1}{15},\frac{1}{12},\frac{1}{10},\ldots\), to \(11\) terms. Answer Q5 2. (i) \(7+10\frac{1}{2}+14+\ldots+84\) Answer Q6 2. (ii) \(34+32+30+\ldots+10\) Answer Q7 2. (iii) \(-5+(-8)+(-11)+\ldots+(-230)\) Answer Q8 3. (i) given \(a=5,\ d=3,\ a_n=50\), find \(n\) and \(S_n\). Answer Q9 3. (ii) given \(a=7,\ a_{13}=35\), find \(d\) and \(S_{13}\). Answer Q10 3. (iii) given \(a_{12}=37,\ d=3\), find \(a\) and \(S_{12}\). Answer Q11 3. (iv) given \(a_3=15,\ S_{10}=125\), find \(d\) and \(a_{10}\). Answer Q12 3. (v) given \(d=5,\ S_9=75\), find \(a\) and \(a_9\). Answer Q13 3. (vi) given \(a=2,\ d=8,\ S_n=90\), find \(n\) and \(a_n\). Answer Q14 3. (vii) given \(a=8,\ a_n=62,\ S_n=210\), find \(n\) and \(d\). Answer Q15 3. (viii) given \(a_n=4,\ d=2,\ S_n=-14\), find \(n\) and \(a\). Answer Q16 3. (ix) given \(a=3,\ n=8,\ S=192\), find \(d\). Answer Q17 3. (x) given \(l=28,\ S=144\), and there are total \(9\) terms. Find \(a\). Answer Q18 4. How many terms of the AP : \(9,17,25,\ldots\) must be taken to give a sum of \(636\)? Answer Q19 5. The first term of an AP is \(5\), the last term is \(45\) and the sum is \(400\). Find the number of terms and the common difference. Answer Q20 6. The first and the last terms of an AP are \(17\) and \(350\) respectively. If the common difference is \(9\), how many terms are there and what is their sum? Answer Q21 7. Find the sum of first \(22\) terms of an AP in which \(d=7\) and \(22\)nd term is \(149\). Answer Q22 8. Find the sum of first \(51\) terms of an AP whose second and third terms are \(14\) and \(18\) respectively. Answer Q23 9. If the sum of first \(7\) terms of an AP is \(49\) and that of \(17\) terms is \(289\), find the sum of first \(n\) terms. Answer Q24 10. (i) Show that \(a_1,a_2,\ldots,a_n,\ldots\) form an AP where \(a_n=3+4n\). Also find the sum of the first \(15\) terms. Answer Q25 10. (ii) Show that \(a_1,a_2,\ldots,a_n,\ldots\) form an AP where \(a_n=9-5n\). Also find the sum of the first \(15\) terms. Answer Q26 11. If the sum of the first \(n\) terms of an AP is \(4n-n^2\), what is the first term (that is \(S_1\))? What is the sum of first two terms? What is the second term? Similarly, find the \(3\)rd, the \(10\)th and the \(n\)th terms. Answer Q27 12. Find the sum of the first \(40\) positive integers divisible by \(6\). Answer Q28 13. Find the sum of the first \(15\) multiples of \(8\). Answer Q29 14. Find the sum of the odd numbers between \(0\) and \(50\). Answer Q30 15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: \(\text{₹}\,200\) for the first day, \(\text{₹}\,250\) for the second day, \(\text{₹}\,300\) for the third day, etc., the penalty for each succeeding day being \(\text{₹}\,50\) more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by \(30\) days? Answer Q31 16. A sum of \(\text{₹}\,700\) is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is \(\text{₹}\,20\) less than its preceding prize, find the value of each of the prizes. Answer Q32 17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant \(1\) tree, a section of Class II will plant \(2\) trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students? Answer Q33 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii \(0.5\text{ cm},1.0\text{ cm},1.5\text{ cm},2.0\text{ cm},\ldots\) as shown in Fig. \(5.4\). What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take \(\pi=\frac{22}{7}\)) Answer Q34 19. \(200\) logs are stacked in the following manner: \(20\) logs in the bottom row, \(19\) in the next row, \(18\) in the row next to it and so on (see Fig. \(5.5\)). In how many rows are the \(200\) logs placed and how many logs are in the top row? Answer Q35 20. In a potato race, a bucket is placed at the starting point, which is \(5\text{ m}\) from the first potato, and the other potatoes are placed \(3\text{ m}\) apart in a straight line. There are ten potatoes in the line (see Fig. \(5.6\)). A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run? Answer Q1 1. Which term of the AP : \(121,117,113,\ldots\), is its first negative term? Answer Q2 2. The sum of the third and the seventh terms of an AP is \(6\) and their product is \(8\). Find the sum of first sixteen terms of the AP. Answer Q3 3. A ladder has rungs \(25\text{ cm}\) apart. (see Fig. \(5.7\)). The rungs decrease uniformly in length from \(45\text{ cm}\) at the bottom to \(25\text{ cm}\) at the top. If the top and the bottom rungs are \(2\frac{1}{2}\text{ m}\) apart, what is the length of the wood required for the rungs? Answer Q4 4. The houses of a row are numbered consecutively from \(1\) to \(49\). Show that there is a value of \(x\) such that the sum of the numbers of the houses preceding the house numbered \(x\) is equal to the sum of the numbers of the houses following it. Find this value of \(x\). Answer Q5 5. A small terrace at a football ground comprises of \(15\) steps each of which is \(50\text{ m}\) long and built of solid concrete. Each step has a rise of \(\frac{1}{4}\text{ m}\) and a tread of \(\frac{1}{2}\text{ m}\). (see Fig. \(5.8\)). Calculate the total volume of concrete required to build the terrace. Answer
The common difference of the AP 2, 5, 8, 11, … is:
A −3 B 2 C 5 D 3
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