Find the best tutors and institutes for Class 10 Tuition
Q3(viii):
In an AP:
(viii) given $a_n = 4, d = 2, S_n = –14$, find $n$ and $a$.
Solution :
Given:
The $n^{th}$ term of the Arithmetic Progression (AP), $a_n = 4$.
The common difference of the AP, $d = 2$.
The sum of the first $n$ terms of the AP, $S_n = -14$.
To find:
The number of terms, $n$, and the first term, $a$.
Step 1: Formulating the equations based on AP formulas.
We use the standard formulas for an Arithmetic Progression:
(i) The $n^{th}$ term formula: $a_n = a + (n - 1)d$
(ii) The sum of $n$ terms formula: $S_n = \frac{n}{2} [a + a_n]$
Step 2: Expressing $a$ in terms of $n$ using the $n^{th}$ term formula.
Substitute the given values $a_n = 4$ and $d = 2$ into the formula $a_n = a + (n - 1)d$:
$4 = a + (n - 1)(2)$
$4 = a + 2n - 2$
$a = 4 - 2n + 2$
$a = 6 - 2n$ --- (Equation 1)
Step 3: Substituting $a$ into the sum formula.
Substitute $S_n = -14$ and $a_n = 4$ into the formula $S_n = \frac{n}{2} [a + a_n]$:
$-14 = \frac{n}{2} [a + 4]$
Multiply both sides by 2:
$-28 = n(a + 4)$ --- (Equation 2)
Step 4: Solving for $n$.
Substitute Equation 1 into Equation 2:
$-28 = n[(6 - 2n) + 4]$
$-28 = n[10 - 2n]$
$-28 = 10n - 2n^2$
Rearrange the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:
$2n^2 - 10n - 28 = 0$
Divide the entire equation by 2 to simplify:
$n^2 - 5n - 14 = 0$
Factorize the quadratic equation by splitting the middle term:
$n^2 - 7n + 2n - 14 = 0$
$n(n - 7) + 2(n - 7) = 0$
$(n - 7)(n + 2) = 0$
This gives two possible values for $n$: $n = 7$ or $n = -2$.
[Since the number of terms $n$ must be a positive integer, we discard $n = -2$.]
Therefore, $n = 7$.
Step 5: Finding the first term $a$.
Substitute $n = 7$ back into Equation 1:
$a = 6 - 2(7)$
$a = 6 - 14$
$a = -8$
Final Answer:
The number of terms $n = 7$ and the first term $a = -8$.
More Questions from Class 10 Mathematics Arithmetic Progression EXERCISE 5.3
- Q1(i): Find the sum of the following APs: (i) 2, 7, 12, . . ., to 10 terms.
- Q1(ii): Find the sum of the following APs: (ii) –37, –33, –29, . . ., to 12 terms.
- Q1(iii): Find the sum of the following APs: (iii) 0.6, 1.7, 2.8, . . ., to 100 terms.
- Q1(iv): Find the sum of the following APs: (iv) $\frac{1}{15}, \frac{1}{12}, \frac{1}{10}$, . . ., to 11 terms.
- Q2(i): Find the sums given below : (i) $7 + 10\frac{1}{2} + 14 + . . . + 84$
- Q2(ii): Find the sums given below : (ii) 34 + 32 + 30 + . . . + 10
- Q2(iii): Find the sums given below : (iii) –5 + (–8) + (–11) + . . . + (–230)
- Q3(i): In an AP: (i) given $a = 5, d = 3, a_n = 50$, find $n$ and $S_n$.
- Q3(ii): In an AP: (ii) given $a = 7, a_{13} = 35$, find $d$ and $S_{13}$.
- Q3(iii): In an AP: (iii) given $a_{12} = 37, d = 3$, find $a$ and $S_{12}$.
- Q3(iv): In an AP: (iv) given $a_3 = 15, S_{10} = 125$, find $d$ and $a_{10}$.
- Q3(ix): In an AP: (ix) given $a = 3, n = 8, S = 192$, find $d$.
- Q3(v): In an AP: (v) given $d = 5, S_9 = 75$, find $a$ and $a_9$.
- Q3(vi): In an AP: (vi) given $a = 2, d = 8, S_n = 90$, find $n$ and $a_n$.
- Q3(vii): In an AP: (vii) given $a = 8, a_n = 62, S_n = 210$, find $n$ and $d$.
- Q3(x): In an AP: (x) given $l = 28, S = 144$, and there are total 9 terms. Find $a$.
- Q4: How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?
- 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.
- 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?
- Q7: Find the sum of first 22 terms of an AP in which $d = 7$ and 22nd term is 149.
- Q8: Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
- 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.
- Q10(i): Show that $a_1, a_2, . . ., a_n, . . .$ form an AP where $a_n$ is defined as below : (i) $a_n = 3 + 4n$ Also find the sum of the first 15 terms in each case.
- Q10(ii): Show that $a_1, a_2, . . ., a_n, . . .$ form an AP where $a_n$ is defined as below : (ii) $a_n = 9 – 5n$ Also find the sum of the first 15 terms in each case.
- 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.
- Q12: Find the sum of the first 40 positive integers divisible by 6.
- Q13: Find the sum of the first 15 multiples of 8.
- Q14: Find the sum of the odd numbers between 0 and 50.
- Q15: A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc., the penalty for each succeeding day being ₹ 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?
- Q16: A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.
- 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?
- 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, . . . as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take $\pi = \frac{22}{7}$) [Hint : Length of successive semicircles is $l_1, l_2, l_3, l_4, . . .$ with centres at A, B, A, B, . . ., respectively.]
- 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?
- 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 $2 \times 5 + 2 \times (5 + 3)$]
CBSE Solutions for Class 10 Mathematics Arithmetic Progression
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Arithmetic Progression
Top Tutor
Kazipet, Warangal
2 yrs of
Exp
1 student
50 hrs of classes completed
Online Classes
Student's Home
Tutor's home
I have 5years of teaching experience where I implement Inductive ,Deductive teaching techinques in mathematics. I begin the teaching concepts of maths with basic level to high level which ensures the teaching learning process occurs even a below average student I make special work sheets for the brain nutrient students and also to the below average students ,so that helps students to become learn individually according to their level. In teaching maths concepts regarding Geometry , Trignometry and Areas and Circles here I use the concrete to semi concrete from thier i take them to abstract so child can easily familiar to the concept, teaching learning process occurs in a positive way. In teaching maths I use heuristic method which creates an enthusiasm among the students to learn maths subject and it eradicate the fear of maths from the students. As a teacher I teach according to the students taste and their interests, so teaching learning process will be occurred in a meaningful way. I observe the individual difference of the students while teaching, so that I can plan to teach according individual difference of the different students in the classroom. I use remedial class for the slow learners to develop in the subject. In this remedial session I mostly concentrating on the basic of maths subject so children can overcome the previous misunderstanding concepts regarding maths and here it will be taught his doubts regarding subject will be clarified. By teaching maths subject I make students all round development which is very valuable in the life.
Top Tutor
Mohkam Pur Kala, Dehradun
8 yrs of
Exp
1 student
Student's Home
Tutor's home
5 years experience in teaching in math and science
Top Tutor
Virupakshapura, Bangalore
19 yrs of
Exp
7 students
427 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I've been teaching class 10 CBSE students for over 12 years now and have improved their ability to grasp concepts faster and implement the concepts through real examples. This method has been useful to ignite a liking for the subject and get better grades.
I'm really grateful to have Rajitha ma'am as my maths teacher. She teaches so well, she explains and clarifies doubts clearly that I'm now able to solve tough sums. The tricks you taught me have helped me in my exams. Thank you so much ma'am for helping me score better in my math exams.
Top Tutor
Nehru Nagar, Belgaum
2 yrs of
Exp
9 students
267 hrs of classes completed
260
per hour
Online Classes
I teach Mathematics to students from Class 7 to Class 10, with 3 years of teaching experience. My focus is on helping students build strong conceptual understanding and develop effective problem-solving skills to excel academically. I have completed my 2nd PUC (Pre-University Course), a Bachelor of Engineering (B.E.) from Visveswaraya Technological University in 2020, and a Bachelor of Education (B.Ed.) from Indira Gandhi National Open University in 2022. I have also completed a program with Cambridge Learning Resources, which has enhanced my teaching techniques and ability to engage students effectively. Over the years, I have guided students to improve their Mathematics understanding and performance, helping them gain confidence in handling both basic and complex problems. My teaching approach emphasizes clarity, practice, and logical reasoning to ensure steady improvement. I have also earned an RPA (Robotic Process Automation) certification in robotics, which reflects my technical skills and commitment to continuous learning. I often incorporate logical thinking and problem-solving strategies inspired by this training into my lessons. I prefer to conduct online classes, offering flexibility in timing and easy access for students. I teach both individual and small group sessions, allowing personalized attention and interactive learning experiences for every student. My classes are interactive, using motivational videos, personal digital pads, and other digital tools to make learning engaging and effective. These resources help students visualize concepts better and retain knowledge more efficiently. I provide custom-prepared notes to support student learning and revision. My ultimate goal is to make Mathematics understandable, enjoyable, and confidence-building, enabling every student to perform at their best and develop a strong foundation for future studies.
This guy is too good at his teaching. He explains the concepts very clear, gets to know about the students, gives his own ideas, own methods, and also gives good advices. He is excellent to consider. I just like him!, he is a great tutor and price is reasonably good.
Super Tutor
Electronic City Phase I, Bangalore
20 yrs of
Exp
24 students
870 hrs of classes completed
400
per hour
Online Classes
Class 10 is a pivotal, fast-paced school year filled with a mix of academic preparation, growing responsibilities, and memorable moments with friends.Daily Routine and StudiesBusy schedules: Days involve attending subject-specific classes, taking detailed notes, and preparing for board exams.Favorite subjects: Time is often spent engaging deeply with core subjects like mathematics, science, and social studies.Extra help: Teachers provide extra guidance and revision sessions to help students handle exam stress.Friendships and Social LifeStronger bonds: Friendships deepen through shared lunch breaks, group studies, and mutual support during exams.Lasting memories: Small moments in the hallway or sharing stories in the classroom create a sense of a second family.
Find more Tutor for Arithmetic Progression in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 10 Mathematics Arithmetic Progression EXERCISE 5.3 worksheets
Download Now
