Find the best tutors and institutes for Class 10 Tuition
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.
Solution :
Given: The $n^{th}$ term of a sequence is defined by the formula $a_n = 9 - 5n$.
To Find:
1. Show that the sequence forms an Arithmetic Progression (AP).
2. Calculate the sum of the first 15 terms ($S_{15}$).
Step 1: Generating the terms of the sequence
To determine if the sequence is an AP, we calculate the first few terms by substituting $n = 1, 2, 3, \dots$ into the given formula $a_n = 9 - 5n$.
For $n = 1$: $a_1 = 9 - 5(1) = 9 - 5 = 4$
For $n = 2$: $a_2 = 9 - 5(2) = 9 - 10 = -1$
For $n = 3$: $a_3 = 9 - 5(3) = 9 - 15 = -6$
For $n = 4$: $a_4 = 9 - 5(4) = 9 - 20 = -11$
Step 2: Verifying the Arithmetic Progression
A sequence is an AP if the difference between consecutive terms is constant. This constant is known as the common difference ($d$).
Calculate the differences:
$d_1 = a_2 - a_1 = -1 - 4 = -5$
$d_2 = a_3 - a_2 = -6 - (-1) = -6 + 1 = -5$
$d_3 = a_4 - a_3 = -11 - (-6) = -11 + 6 = -5$
[Since $a_{n} - a_{n-1} = -5$ for all $n$, the difference is constant.]
Therefore, the sequence $4, -1, -6, -11, \dots$ forms an AP with the first term $a = 4$ and common difference $d = -5$.
Step 3: Calculating the sum of the first 15 terms
The formula for the sum of the first $n$ terms of an AP is given by:
$S_n = \frac{n}{2} [2a + (n - 1)d]$
Here, $n = 15$, $a = 4$, and $d = -5$. Substituting these values into the formula:
$S_{15} = \frac{15}{2} [2(4) + (15 - 1)(-5)]$
$S_{15} = \frac{15}{2} [8 + (14)(-5)]$
$S_{15} = \frac{15}{2} [8 - 70]$
$S_{15} = \frac{15}{2} [-62]$
Step 4: Final Arithmetic Simplification
$S_{15} = 15 \times \left(\frac{-62}{2}\right)$
$S_{15} = 15 \times (-31)$
$S_{15} = -465$
Final Answer: The sequence forms an AP with a common difference of $-5$, and the sum of the first 15 terms is $-465$.
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.
- 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.
- 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?
- 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)
- 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 $2 \times 5 + 2 \times (5 + 3)$]
- 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(viii): In an AP: (viii) given $a_n = 4, d = 2, S_n = –14$, find $n$ and $a$.
- 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.
CBSE Solutions for Class 10 Mathematics Arithmetic Progression
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Arithmetic Progression
Top Tutor
West Marredpally, Hyderabad
20 yrs of
Exp
92 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have more than 15 years teaching experience in teaching CBSE classes 8-10 Physics/Chemistry. I can also teach math upto 9th . All my students secured 10/10 in previous years exams. And in school more than 90% students got more than 90 in Science Exam and all have passed in Science i. Physics and Chemistry.
Top Tutor
Thane West, Thane
16 yrs of
Exp
17 students
3973 hrs of classes completed
350
per hour
Online Classes
This is Debabrato Chatterjee, an Maths and Science tutor with more than fifteen years of experience. I have been and am teaching students of IB, IGCSE, CBSE, ICSE, and Maharastra State Boards. My teaching style is to learning from concepts and their application. Solving many papers and completing homework helps grasp the concepts. I teach students from grade VI to XII. Please let me know when you would like to schedule a demo.
Super Tutor
Itagalpura Main Rd, Madappanahalli
17 yrs of
Exp
133 students
3628 hrs of classes completed
500
per hour
Online Classes
I gained extensive experience in delivering high-quality education following the CBSE/ICSE curriculum. My key responsibilities included: Classroom Teaching: Effectively taught Mathematics, Physics and Chemistry to students of classesX, ensuring alignment with the CBSE/ ICSE syllabus and fostering a deep understanding of key concepts. Curriculum Planning: Designed and implemented lesson plans that adhered to the CBSE guidelines, ensuring comprehensive coverage of topics while keeping in mind the varying learning needs of students. Student Assessment: Created and administered tests, quizzes, and assignments in accordance with the CBSE exam pattern. Monitored student performance and provided constructive feedback to facilitate continuous improvement. Interactive Learning: Encouraged an interactive classroom environment, integrating various teaching aids such as multimedia presentations, hands-on activities, and group discussions to enhance student engagement and understanding. Preparation for Board Exams: Provided focused support for students appearing for their CBSE/ICSE board exams by conducting regular revision sessions, mock tests, and personalized guidance to help them excel. Parental Communication: Maintained regular communication with parents regarding student progress, attendance, and overall development. Organized parent-teacher meetings to ensure a collaborative approach to student growth. Co-curricular Involvement: Actively participated in school events, projects, and extracurricular activities, fostering holistic development among students. Professional Development: Engaged in continuous professional development through workshops and CBSE training programs to stay updated with the latest educational practices and policies.
I have approached Sai sir for my daughter Maths tuition. The way you are interacting with student making learning so interesting and explaining everything was just outstanding. Your support and encouragement have given me the confidence.Thank you so much sir for your patience. Your teaching skills are very unique so that any student can easily grasp and enjoy learning.
Super Tutor
Kharghar Sector 27, Mumbai
13 yrs of
Exp
30 students
1081 hrs of classes completed
650
per hour
Online Classes
Student's Home
**Teaching Methodology:** My approach to teaching Mathematics and Science is personalized and interactive. I focus on understanding each student's unique learning style and adapt my methods accordingly. My goal is to make Maths and Science relatable and engaging, using real-life examples and practical applications to enhance understanding.I try to develop self thinking on problem and pushing them to solve by self **Key Skills:** - Strong grasp of Mathematics concepts across various levels - Ability to simplify complex topics for better understanding - Excellent communication and motivational skills - Patient and supportive, ensuring a comfortable learning environment - Regular progress assessments and constructive feedback **Services Offered:** - One-on-one personalized tutoring sessions - Homework and assignment assistance - Exam preparation and revision classes - Online and in-person tutoring options - Flexible scheduling to accommodate student needs **Achievements:** - Helped students improve their grades by an average of [25%] - Successfully prepared students for competitive exams with [JEE,NEET,IMO] - Received positive feedback and high ratings from students and parents Interested in improving your Maths AND Science skills? Contact me today to schedule a session and take the first step towards mastering Mathematics and Science!
This teacher is incredibly friendly and teaches in a very calm and patient manner. He helped me clear all my doubts; after failing my board exams previously, I managed to score 47/80 in Mathematics under his guidance. He truly excels at building a student's confidence.
Top Tutor
Tilak Nagar, Delhi
4 yrs of
Exp
2 students
56 hrs of classes completed
300
per hour
Online Classes
Student's Home
Tutor's home
I am a mathematics expert . I am teaching this subject since a decade . I ve handled thousands of students and bring numerous centum results in past . You will able to learn maths in the most pragmatic approach with me and develop a love for this subject .
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
