--- title: "NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.2" url: https://www.swavid.com/maths/class/9/chapter/predicting-what-comes-next-exploring-sequences-and-progressions/ncert-solutions/exercise-8-2 dateModified: 2026-10-07T15:49:15+00:00 --- # NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.2 Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions. Every question from Exercise 8.2, with full working and the final answer. Free PDF (21 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-9/swavid-ncert-solutions-class-9-maths-chapter-8-predicting-what-comes-next-exploring-sequences-and-progressions-13ea0a2ffd.pdf ## Exercise Set 8.2 ### Question 1 *2 marks · Very short answer* Find the $10^{\text{th}}$ and $26^{\text{th}}$ terms of the AP: 3, 8, 13, 18, ... **Solution** 1. Given AP is 3, 8, 13, 18, ..., with first term $a = 3$ and common difference $d = 8 - 3 = 5$. 2. Using the formula $t_n = a + (n - 1)d$, the 10th term is $t_{10} = 3 + (10 - 1) \times 5 = 3 + 45 = 48$ and the 26th term is $t_{26} = 3 + (26 - 1) \times 5 = 3 + 125 = 128$. **Answer:** The 10th term is 48 and the 26th term is 128. > Common mistake: Using $n$ instead of $n-1$ in the formula. ### Question 2 *3 marks · Short answer* Which term of the AP: 21, 18, 15, ... is $-81$? Also, is 0 a term of this AP? Give reasons for your answer. **Solution** 1. Given AP is 21, 18, 15, ..., with $a = 21$ and $d = 18 - 21 = -3$. 2. To find which term is $-81$, we set $t_n = -81$, giving $21 + (n - 1)(-3) = -81$, which simplifies to $-3(n - 1) = -102$, so $n - 1 = 34$ and $n = 35$. Thus, $-81$ is the 35th term. 3. To check if 0 is a term, we set $t_n = 0$, giving $21 + (n - 1)(-3) = 0$, which leads to $3(n - 1) = 21$, so $n - 1 = 7$ and $n = 8$. Since $n$ is a natural number, 0 is the 8th term of this AP. **Answer:** -81 is the 35th term, and 0 is the 8th term of the AP. > Common mistake: Treating non-integer values of $n$ as valid term positions. ### Question 3 *3 marks · Short answer* Find the $n^{\text{th}}$ term of the AP: 11, 8, 5, 2, ... Write the recursive rule for this AP. **Solution** 1. Given AP is 11, 8, 5, 2, ..., with first term $a = 11$ and common difference $d = 8 - 11 = -3$. 2. The $n^{\text{th}}$ term is given by $t_n = a + (n - 1)d = 11 + (n - 1)(-3) = 11 - 3n + 3 = 14 - 3n$. 3. The recursive rule for this AP is $t_1 = 11$ and $t_n = t_{n - 1} - 3$ for $n \ge 2$. **Answer:** The $n^{\text{th}}$ term is $14 - 3n$ and the recursive rule is $t_1 = 11, t_n = t_{n-1} - 3$ for $n \ge 2$. > Common mistake: Incorrect sign when finding the common difference $d$. ### Question 4 *3 marks · Short answer* An AP consists of 50 terms in which the $3^{\text{rd}}$ term is 12 and the last term is 106. Find the $29^{\text{th}}$ term. (Hint: If '$a$' is the first term and '$d$' the common difference, then we arrive at the equations $a + 2d = 12$ and $a + 49d = 106$. Solve this pair of linear equations for '$a$' and '$d$'.) **Solution** 1. Let $a$ be the first term and $d$ be the common difference. Given that the 3rd term is 12, we have $a + 2d = 12$. 2. The last term of 50 terms is 106, so the 50th term is $a + 49d = 106$. 3. Subtracting the first equation from the second gives $47d = 94$, so $d = 2$. Substituting $d = 2$ gives $a = 12 - 4 = 8$. 4. The 29th term is $t_{29} = a + 28d = 8 + 28(2) = 8 + 56 = 64$. **Answer:** The 29th term of the AP is 64. > Common mistake: Arithmetic errors while solving simultaneous equations. ### Question 5 *3 marks · Short answer* How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers? **Solution** 1. The two-digit numbers divisible by 3 form the AP: 12, 15, 18, ..., 99, with $a = 12$, $d = 3$, and $t_n = 99$. 2. Using $t_n = a + (n - 1)d$, we get $12 + (n - 1)3 = 99$, which gives $3(n - 1) = 87$, so $n - 1 = 29$ and $n = 30$ numbers. 3. The sum of these $n$ numbers can be found using the formula $S_n = \frac{n}{2}(a + l) = \frac{30}{2}(12 + 99) = 15 \times 111 = 1665$. **Answer:** There are 30 such 2-digit numbers and their sum is 1665. > Common mistake: Incorrect identification of the first and last two-digit multiples of 3. ### Question 6 *3 marks · Short answer* Harish started work at an annual salary of `5,00,000 and received an increment of `20,000 each year. After how many years did his income reach `7,00,000? **Solution** 1. The annual salaries form an AP with first term $a = 5,00,000$ and common difference $d = 20,000$. 2. The explicit formula for the $n^{\text{th}}$ term is given by $t_n = a + (n - 1)d$. 3. Substituting the values, we get $7,00,000 = 5,00,000 + (n - 1) \times 20,000$. 4. Solving for $n$, we get $2,00,000 = (n - 1) \times 20,000$, which gives $n - 1 = 10$, so $n = 11$. 5. Thus, his income reached $7,00,000$ in the 11th year, which is after 10 years of receiving increments. **Answer:** After 10 years (in the 11th year) > Common mistake: Students often write 11 years instead of 10 years of service, confusing the term number $n = 11$ with the number of elapsed years. ### Question 7 *3 marks · Short answer* A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all? **Solution** 1. The number of marbles in each row forms the sequence $1, 2, 3, \dots, 25$, which is the sequence of natural numbers. 2. The total number of marbles used is the sum of the first 25 natural numbers, given by the formula $S_n = \frac{n(n + 1)}{2}$. 3. Substituting $n = 25$, we get $S_{25} = \frac{25(25 + 1)}{2} = \frac{25 \times 26}{2} = 25 \times 13 = 325$. **Answer:** 325 marbles > Common mistake: Multiplying the middle term incorrectly or forgetting to divide by 2 in the sum formula. ## Related pages - [All Chapter 8 NCERT solutions](https://www.swavid.com/maths/class/9/chapter/predicting-what-comes-next-exploring-sequences-and-progressions/ncert-solutions) - [Exercise 8.1](https://www.swavid.com/maths/class/9/chapter/predicting-what-comes-next-exploring-sequences-and-progressions/ncert-solutions/exercise-8-1) - [Exercise 8.3](https://www.swavid.com/maths/class/9/chapter/predicting-what-comes-next-exploring-sequences-and-progressions/ncert-solutions/exercise-8-3) Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student