Chapter 5 · Arithmetic Progressions

NCERT Exemplar Class 10 Maths Exercise 5.2

All 7 questions of NCERT Exemplar Class 10 Maths Exercise 5.2 (Arithmetic Progressions), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Which of ​the following form an ​AP? ​Justify your ​answer. ​

   1. (i)$-1,-1,-1,-1,\dots$
   2. (ii)$0,2,0,2,\dots$
   3. (iii)$1,1,2,2,3,3,\dots$
   4. (iv)$11,22,33,\dots$ ​
   5. (v)$\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ ​
   6. (vi)$2,2^2,2^3,2^4,\dots$
   7. (vii)$\sqrt{3},\sqrt{12},\sqrt{27},\sqrt{48},\dots$
   Show answer

   (i) AP ($d=0$); (ii) Not an AP; (iii) Not an AP; (iv) AP ($d=11$); (v) Not an AP; (vi) Not an AP (it is a GP); (vii) AP ($d=\sqrt{3}$).

   Step-by-step solution →
2. Q3
   For ​the AP: $-3,-7,-11,\dots$, can we find directly $a_{30}-a_{20}$ without actually ​finding $a_{30}$ and $a_{20}$? ​Give reasons ​for ​your ​answer.
   Show answer

   Yes - $a_{30}-a_{20}=10d=10(-4)=-40$, obtained directly because $a$ cancels.

   Step-by-step solution →
3. Q4
   Two APs have the same ​common difference. The first term of one ​AP is 2 and that of ​the other ​is ​7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the ​same as the difference between any two corresponding terms. Why?
   Show answer

   Because both APs have the same common difference d, the term A_n - a_n = [7 + (n-1)d] - [2 + (n-1)d] = 5 for every n. The (n-1)d parts cancel, so the difference of corresponding terms is always 5, the difference of the first terms.

   Step-by-step solution →
4. Q5
   Is $0$ a term ​of ​the ​AP: $31,28,25,\dots$? ​Justify ​your ​answer.
   Show answer

   No. Setting $34-3n=0$ gives $n=\frac{34}{3}$, which is not a positive integer, so $0$ is not a term of the AP.

   Step-by-step solution →
5. Q6
   The taxi fare after each km, when the fare is Rs 15 for the ​first km and Rs ​8 ​for each additional ​km, does not form an AP as the ​total fare (in Rs) after ​each km ​is 15, 8, 8, 8, ... Is the statement ​true? Give reasons.
   Show answer

   The statement is false. The correct total-fare sequence is $15,23,31,39,\dots$, which **is** an AP with first term 15 and common difference 8.

   Step-by-step solution →
6. Q7
   In which of the following situations, ​do the lists of numbers involved form an AP? ​Give reasons for your answers.

   1. (i)The fee charged from a student every month by a school for the whole session, when the monthly fee is Rs 400. ​
   2. (ii)​The fee charged every month by a school from Classes I to XII, when the monthly ​fee for Class I is Rs ​250, and it increases by Rs 50 for ​the next higher class.
   3. (iii)The amount of money in the account of Varun at the end of every year when Rs 1000 is deposited at simple interest of 10% per annum.
   4. (iv)The number of bacteria ​in a certain food item after each second, when they double in every second.

   1. (i)Rs 400 monthly fee each month
   2. (ii)Class I fee Rs 250, increases Rs 50 per class
   3. (iii)Rs 1000 at 10% simple interest, yearly amount
   4. (iv)Bacteria doubling every second
   Show answer

   (i), (ii) And (iii) form APs (common differences $0$, $50$, $100$); (iv) Does not - it is a GP with ratio $2$.

   Step-by-step solution →
7. Q8
   Justify ​whether ​it ​is true to ​say that the following ​are ​the $n^{\text{th}}$ terms of an ​AP.

   1. (i)$2n-3$
   2. (ii)$3n^2+5$
   3. (iii)$1+n+n^2$
   Show answer

   (i) $2n-3$ **is** an AP (common difference 2). (ii) $3n^2+5$ is **not** an AP. (iii) $1+n+n^2$ is **not** an AP.

   Step-by-step solution →