Chapter 5 · Arithmetic Progressions

NCERT Class 10 Maths Exercise 5.3

All 20 questions of NCERT Class 10 Maths Exercise 5.3 (Arithmetic Progressions), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Find the sum of the following APs: ​

   1. (i)​2, 7, ​12, ..., to 10 terms.
   2. (ii)-37, -33, -29, ..., to 12 ​terms.
   3. (iii)​0.6, 1.7, 2.8, ..., ​to 100 terms.
   4. (iv)1/15, 1/12, ​1/10, ​..., to 11 terms.
   Show answer

   (i) $245$; (ii) $-180$; (iii) $5505$; (iv) $\frac{33}{20}$.

   Step-by-step solution →
2. Q2
   Find ​the sums given ​below: ​

   1. (i)$7+10\frac{1}{2}+14+\dots +84$
   2. (ii)$34+32+30+\dots +10$ ​
   3. (iii)$-5+(-8)+(-11)+\dots +(-230)$
   Show answer

   (i) $S_{23}=\frac{2093}{2}=1046\frac{1}{2}$; (ii) $S_{13}=286$; (iii) $S_{76}=-8930$.

   Step-by-step solution →
3. Q3
   In ​an ​AP:

   1. (i)given $a=5$, $d=3$, $a_n=50$, find $n$ and $S_n$.
   2. (ii)given $a=7$, $a_{13}=35$, find $d$ and $S_{13}$.
   3. (iii)given $a_{12}=37$, $d=3$, find $a$ ​and $S_{12}$.
   4. (iv)given $a_3=15$, $S_{10}=125$, find $d$ and $a_{10}$.
   5. (v)given $d=5$, $S_9=75$, find $a$ and $a_9$.
   6. (vi)given $a=2$, $d=8$, $S_n=90$, ​find $n$ and $a_n$.
   7. (vii)​given $a=8$, $a_n=62$, $S_n=210$, find $n$ and $d$.
   8. (viii)given $a_n=4$, $d=2$, $S_n=-14$, ​find $n$ and $a$.
   9. (ix)given $a=3$, $n=8$, $S=192$, find $d$.
   10. (x)​given $l=28$, $S=144$, $n=9$, find $a$.
   Show answer

   (i) $n=16,S_{16}=440$; (ii) $d=\frac{7}{3},S_{13}=273$; (iii) $a=4,S_{12}=246$; (iv) $d=-1,a_{10}=8$; (v) $a=-\frac{35}{3},a_9=\frac{85}{3}$; (vi) $n=5,a_n=34$; (vii) $n=6,d=\frac{54}{5}$; (viii) $n=7,a=-8$; (ix) $d=6$; (x) $a=4$.

   Step-by-step solution →
4. Q4
   How ​many terms of the AP : $9,17,25,\dots$ must ​be ​taken ​to ​give a sum ​of ​636?
   Show answer

   $n=12$ terms are needed.

   Step-by-step solution →
5. Q5
   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.
   Show answer

   $n=16$ and $d=\frac{8}{3}$.

   Step-by-step solution →
6. Q6
   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?
   Show answer

   The AP has $n=38$ terms and their sum is $S_{38}=6973$.

   Step-by-step solution →
7. Q7
   Find the sum of first ​22 terms ​of ​an ​AP in ​which $d=7$ and ​22nd term is 149.
   Show answer

   $S_{22}=1661$.

   Step-by-step solution →
8. Q8
   Find ​the sum of first $51$ terms ​of ​an AP ​whose ​second and ​third terms are $14$ ​and $18$ respectively.
   Show answer

   $S_{51}=5610$.

   Step-by-step solution →
9. Q9
   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.
   Show answer

   $S_n=n^2$ (with $a=1$, $d=2$).

   Step-by-step solution →
10. Q10
    Show ​that $a_1,a_2,\dots ,a_n,\dots$ form ​an AP where $a_n$ is ​defined as below. Also find ​the sum ​of ​the first 15 terms ​in each case.

    1. (i)$a_n=3+4n$
    2. (ii)$a_n=9-5n$
    Show answer

    (i) AP with $a=7,d=4$ and $S_{15}=525$. (ii) AP with $a=4,d=-5$ and $S_{15}=-465$.

    Step-by-step solution →
11. Q11
    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 3rd, the 10th and ​the $n$th terms.
    Show answer

    $a_1=3$, $S_2=4$, $a_2=1$, $a_3=-1$, $a_{10}=-15$, and the general term $a_n=5-2n$.

    Step-by-step solution →
12. Q12
    Find ​the sum ​of the first ​40 ​positive ​integers divisible ​by ​6.
    Show answer

    $S_{40}=4920$.

    Step-by-step solution →
13. Q13
    Find ​the ​sum ​of ​the ​first $15$ multiples ​of $8$.
    Show answer

    $S_{15}=960$.

    Step-by-step solution →
14. Q14
    Find ​the sum of ​the ​odd numbers ​between 0 ​and 50.
    Show answer

    The sum of the odd numbers between $0$ and $50$ is $S_{25}=625$.

    Step-by-step solution →
15. Q15
    A contract on construction job ​specifies a penalty for delay of completion beyond a ​certain date as follows: Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the ​third day, ​etc., the penalty ​for each succeeding day being Rs 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?
    Show answer

    The contractor has to pay a total penalty of Rs $27,750$.

    Step-by-step solution →
16. Q16
    A sum of $\text{Rs }700$ is ​to be used to give ​seven cash prizes to students of ​a ​school for their overall academic performance. If each prize is $\text{Rs }20$ less than ​its preceding ​prize, find the value ​of each ​of the prizes.
    Show answer

    With $a=160$ and $d=-20$, the seven prizes (in Rs) are $160,140,120,100,80,60,40$.

    Step-by-step solution →
17. Q17
    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?
    Show answer

    The students will plant a total of $234$ trees.

    Step-by-step solution →
18. Q18
    A spiral is made up of successive semicircles, with centres alternately ​at $A$ and $B$, starting with centre at $A$, of radii $0.5$ cm, $1.0$ ​cm, $1.5$ cm, $2.0$ cm, $\dots$ as shown in Fig. $5.4$. What ​is ​the total length ​of such a spiral made up ​of ​thirteen consecutive ​semicircles? (Take $\pi =\frac{22}{7}$.)

    Show answer

    The total length of the spiral is $143$ cm.

    Step-by-step solution →
19. Q19
    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?

    Show answer

    The $200$ logs are placed in $16$ rows, and the top row contains $5$ logs (the root $n=25$ is rejected because it would make the top row hold $-4$ logs).

    Step-by-step solution →
20. Q20
    In a potato race, a ​bucket is ​placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 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? [Hint: To pick up the first potato and the second potato, the total distance (in metres) run by a competitor is $2\times 5+2\times (5+3)$.]

    Show answer

    The competitor runs a total distance of $370$ m.

    Step-by-step solution →