Chapter 5 · Arithmetic Progressions

NCERT Exemplar Class 10 Maths Exercise 5.3

All 35 questions of NCERT Exemplar 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
   Match ​the APs given ​in column ​A ​with suitable ​common ​differences given in column ​B.
   Show answer

   Matching: A1-B4 ($d=-4$), A2-B5 ($d=2$), A3-B1 ($d=2/3$), A4-B2 ($d=-5$).

   Step-by-step solution →
2. Q2
   Verify ​that each of the ​following ​is an ​AP, and then ​write ​its next three terms. ​

   1. (i)$0,\frac{1}{4},\frac{1}{2},\frac{3}{4},\dots$
   2. (ii)$5,\frac{14}{3},\frac{13}{3},4,\dots$
   3. (iii)$\sqrt{3},2\sqrt{3},3\sqrt{3},\dots$
   4. (iv)$a+b,(a+1)+b,(a+1)+(b+1),\dots$
   5. (v)$a,2a+1,3a+2,4a+3,\dots$
   Show answer

   All five lists are APs. Next three terms: (i) $1,\frac{5}{4},\frac{3}{2}$; (ii) $\frac{11}{3},\frac{10}{3},3$; (iii) $4\sqrt{3},5\sqrt{3},6\sqrt{3}$; (iv) $a+b+3,a+b+4,a+b+5$; (v) $5a+4,6a+5,7a+6$.

   Step-by-step solution →
3. Q3
   Write the first three ​terms ​of ​the ​APs ​when $a$ and $d$ ​are as given below:

   1. (i)$a=\frac{1}{2},d=-\frac{1}{6}$ ​
   2. (ii)$a=-5,d=-3$
   3. (iii)$a=\sqrt{2},d=\frac{1}{\sqrt{2}}$
   Show answer

   (i) $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ (ii) $-5,-8,-11$ (iii) $\sqrt{2},\frac{3}{\sqrt{2}},2\sqrt{2}$.

   Step-by-step solution →
4. Q4
   Find $a$, $b$ and $c$ such ​that ​the ​following ​numbers ​are ​in ​AP: $a,7,b,23,c$.
   Show answer

   $a=-1$, $b=15$, $c=31$.

   Step-by-step solution →
5. Q5
   Determine the ​AP ​whose fifth term is 19 ​and the difference ​of the eighth ​term from the thirteenth ​term is 20.
   Show answer

   The AP is 3, 7, 11, 15, 19, ... With first term 3 and common difference 4.

   Step-by-step solution →
6. Q6
   The 26th, 11th ​and the last term of ​an ​AP are 0, 3 and $-\frac{1}{5}$, respectively. Find the common ​difference and the ​number of ​terms.
   Show answer

   Common difference $d=-\frac{1}{5}$ and the number of terms $n=27$ (with first term $a=5$).

   Step-by-step solution →
7. Q7
   The sum ​of the ​5th and the 7th terms of an ​AP ​is 52 ​and the 10th ​term is 46. Find the AP.
   Show answer

   The AP is $1,6,11,16,21,\dots$ with first term $a=1$ and common difference $d=5$.

   Step-by-step solution →
8. Q8
   Find ​the ​20th term ​of the AP whose 7th term is ​24 less than ​the 11th ​term, first term ​being 12.
   Show answer

   The 20th term is $a_{20}=126$.

   Step-by-step solution →
9. Q9
   If the $9^{th}$ term ​of an AP ​is ​zero, prove ​that ​its $29^{th}$ ​term ​is ​twice its $19^{th}$ term.
   Show answer

   Proved: with $a_9=0$ we get $a=-8d$, so $a_{19}=10d$ and $a_{29}=20d=2a_{19}$.

   Step-by-step solution →
10. Q10
    Find whether ​55 ​is a term ​of the AP: 7, 10, ​13, ... or ​not. If yes, find which ​term it ​is.
    Show answer

    Yes, 55 is a term of the AP -- it is the 17th term.

    Step-by-step solution →
11. Q11
    Determine $k$ so ​that $k^2+4k+8$, $2k^2+3k+6$, $3k^2+4k+4$ ​are ​three ​consecutive ​terms of ​an ​AP.
    Show answer

    $k=0$; the three terms become $8$, $6$, $4$ with common difference $-2$.

    Step-by-step solution →
12. Q12
    Split ​207 ​into three parts such ​that these ​are in AP ​and the product ​of the two smaller parts ​is 4623.
    Show answer

    The three parts are 67, 69 and 71 (an AP with common difference 2).

    Step-by-step solution →
13. Q13
    The angles of a triangle are in AP. ​The ​greatest angle ​is twice ​the least. Find ​all the angles of the ​triangle.
    Show answer

    The angles are $40^{\circ },60^{\circ }$ and $80^{\circ }$.

    Step-by-step solution →
14. Q14
    If the $n$th terms of the ​two ​APs: $9,7,5,\dots$ and $24,21,18,\dots$ are ​the ​same, find the ​value of $n$. ​Also ​find ​that term.
    Show answer

    $n=16$, and the common term is $-21$.

    Step-by-step solution →
15. Q15
    If sum ​of the 3rd ​and ​the 8th terms ​of an AP is 7 and the sum ​of the 7th and ​the 14th terms ​is $-3$, find the 10th term.
    Show answer

    The 10th term is $a_{10}=-1$ (with $a=8$, $d=-1$).

    Step-by-step solution →
16. Q16
    Find ​the $12^{\text{th}}$ ​term ​from the ​end ​of ​the AP: $-2,-4,-6,\dots ,-100$.
    Show answer

    The 12th term from the end is $-78$.

    Step-by-step solution →
17. Q17
    Which term ​of the AP: 53, ​48, 43, ​... ​is the first negative ​term?
    Show answer

    The 12th term is the first negative term ($a_{12}=-2$).

    Step-by-step solution →
18. Q18
    How many numbers lie between ​10 and 300, which when ​divided ​by 4 ​leave a remainder ​3?
    Show answer

    There are $73$ such numbers.

    Step-by-step solution →
19. Q19
    Find ​the ​sum of ​the two middle most ​terms of ​the ​AP: $-\frac{4}{3},-1,-\frac{2}{3},\dots ,4\frac{1}{3}$.
    Show answer

    The sum of the two middle-most terms is $3$.

    Step-by-step solution →
20. Q20
    The first ​term of ​an AP is $-5$ and the ​last term is $45$. If the sum of ​the terms ​of the AP is $120$, then find the number ​of terms and ​the common difference.
    Show answer

    The AP has $n=6$ terms with common difference $d=10$.

    Step-by-step solution →
21. Q21
    Find the sum: ​

    1. (i)$1+(-2)+(-5)+(-8)+\dots +(-236)$ ​
    2. (ii)$\left(4-\frac{1}{n}\right)+\left(4-\frac{2}{n}\right)+\left(4-\frac{3}{n}\right)+\dots$ upto $n$ ​terms ​
    3. (iii)$\frac{a-b}{a+b}+\frac{3a-2b}{a+b}+\frac{5a-3b}{a+b}+\dots$ ​to 11 terms.
    Show answer

    (i) $-9400$; (ii) $\frac{7n-1}{2}$; (iii) $\frac{11(11a-6b)}{a+b}$.

    Step-by-step solution →
22. Q22
    Which ​term of ​the AP: $-2,-7,-12,\dots$ will be $-77$? Find ​the sum of this ​AP upto the term $-77$.
    Show answer

    $-77$ is the $16$th term, and the sum up to it is $S_{16}=-632$.

    Step-by-step solution →
23. Q23
    If $a_n=3-4n$, show that $a_1,a_2,a_3,\dots$ ​form ​an AP. ​Also find $S_{20}$.
    Show answer

    The sequence is an AP with $a=-1$, $d=-4$; $S_{20}=-780$.

    Step-by-step solution →
24. Q24
    In ​an ​AP, ​if $S_n=n(4n+1)$, ​find ​the AP.
    Show answer

    The AP is $5,13,21,29,\dots$ with first term $5$ and common difference $8$ ($a_n=8n-3$).

    Step-by-step solution →
25. Q25
    In ​an ​AP, if $S_n=3n^2+5n$ ​and $a_k=164$, ​find the value of $k$.
    Show answer

    $k=27$.

    Step-by-step solution →
26. Q26
    If $S_n$ denotes the sum ​of ​first $n$ ​terms ​of an ​AP, ​prove ​that $S_{12}=3(S_8-S_4)$.
    Show answer

    Both sides equal $12a+66d$, hence $S_{12}=3(S_8-S_4)$ is proved.

    Step-by-step solution →
27. Q27
    Find the sum ​of first 17 ​terms ​of an ​AP whose $4^{th}$ ​and $9^{th}$ terms are $-15$ ​and $-30$ ​respectively.
    Show answer

    With $a=-6$ and $d=-3$, $S_{17}=\frac{17}{2}(-60)=-510$.

    Step-by-step solution →
28. Q28
    If sum of first 6 terms of an ​AP is 36 and that of ​the first 16 terms is 256, ​find ​the sum of first 10 ​terms.
    Show answer

    The sum of the first 10 terms is $S_{10}=100$ (with $a=1$, $d=2$).

    Step-by-step solution →
29. Q29
    Find ​the sum of ​all the 11 ​terms of an ​AP whose ​middle ​most ​term ​is 30.
    Show answer

    The sum of all 11 terms is $330$.

    Step-by-step solution →
30. Q30
    Find the ​sum of ​last ten ​terms ​of the ​AP: $8,10,12,\dots ,126$.
    Show answer

    The sum of the last ten terms is $1170$.

    Step-by-step solution →
31. Q31
    Find ​the sum of first ​seven numbers ​which ​are multiples of 2 ​as well as of 9. ​[Hint: Take the LCM ​of 2 ​and 9.]
    Show answer

    The seven numbers are $18,36,54,72,90,108,126$ and their sum is $S_7=504$.

    Step-by-step solution →
32. Q32
    How ​many ​terms of the AP: $-15,-13,-11,\dots$ are ​needed ​to ​make the ​sum $-55$? ​Explain the reason for double answer.
    Show answer

    $n=5$ or $n=11$. The double answer occurs because terms $a_6$ to $a_{11}$ (namely $-5,-3,-1,1,3,5$) sum to $0$, so $S_5=S_{11}=-55$.

    Step-by-step solution →
33. Q33
    The sum of the first $n$ terms ​of an AP whose first term is 8 and the common difference is ​20 is equal to ​the sum of first $2n$ terms ​of another AP ​whose first term is $-30$ ​and the common difference is ​8. Find $n$.
    Show answer

    $n=11$.

    Step-by-step solution →
34. Q34
    Kanika ​was given her pocket money on Jan $1^{st}$, $2008$. ​She puts Re $1$ on Day $1$, Rs $2$ on Day $2$, Rs $3$ on Day $3$, and continued doing so till the end of the month, from this money into ​her piggy bank. She also spent Rs $204$ of her pocket money, and found that at the end of ​the ​month she still ​had Rs $100$ with her. How much was ​her pocket money for the month?
    Show answer

    Kanika's pocket money for the month was Rs $800$.

    Step-by-step solution →
35. Q35
    Yasmeen ​saves Rs $32$ during the ​first month, Rs $36$ ​in ​the second month and Rs $40$ ​in the third month. If she continues to ​save in this manner, in how many months will she ​save Rs $2000$?
    Show answer

    Yasmeen will save Rs $2000$ in $25$ months.

    Step-by-step solution →