Finding nth term of sequences

An Action Research on Problem in finding the nth term of a sequence

By : Krishna Singh Pela

Instructor of Mathematics

Educational Training Center,"B"

Dhangadhi, Kailali, Nepal

Introduction

There are many types of sequences and series i.e. arithmetic sequence, geometric sequence where there is a certain pattern and the difference of two consecutive terms or the ratio of two consecutive terms is always equal or constant. In such sequences, we can easily find the nth term or the summation of the series by using the formulae. But there are some complicated sequences where there is neither the first difference nor the first ratio is constant. And if we try to find the nth term of such sequences, we can't get it easily. There are two options :

 Use inspection/induction method and try until the result is found

 Use some special formula and find it.

Statement of the problem
The problem is to find the general term of a given sequence or we can say it the nth term of the sequence.

Objective of the study

The main objective this study is to call the demand of trainee teachers of TPD (2070), of Kailali district and to make them able to teach this topic in their classroom. This is a general type of problem so the objective of this study is also focused towards preparing sufficient reference materials on this topic so that I will feel easy in my future training classes.

Significance of the study

 It's a textbook based problem of high school mathematics(optional math.)

 The high school teachers have been facing a great problem on this topic. When they fill need proposal form for TPD, they put this problem each time.

 There is no sufficient reference given in the textbook and the method is based on induction.

Data collection

List of teachers who put the demand

S.N.	Name of Teachers	Name of School
1	Puskar Raj Bhatta	Saraswati HSS, Geta
2	Govind Prasad Bhatta	Sharada HSS, Baiyabehedi
3	Bikalpa Chaudhari	Laxmi HSS, Udasipur
4	Asharam Chaudhari	Mahendra Shishu HSS, Dharmapur
5	Rajendra Pokharel	Khadgasmriti HSS, Tikapur
6	Jang Bahadur Bam	Bhanu ma.vi. Pathari
7	Harendraraj Awasthi	Janata Ra.ma.vi. Kailali gaun
8	Gyanendraraj Joshi	Trinagar HSS, Dhangadhi
9	Uddhav Dev Bhatt	Jankalyan HSS, Jkugeda
10	Janak Prasad Bhatt	Phulbari HSS, Phulbari
11	Karna Bahadur Kadayat	Gwasi HSS, Malakheti
12	Gorakh Nath	Shaileswori HSS, Badeha
13	Radha Bisht	Saraswati HSS, Pratappur
14	Raj Bahadur Kadayat	Siddsha Baba Ma.vi. Geti
15	Ganesh Datt Joshi	Panchoday HSS, Dhangadhi
16	Devendra Jha	Trinagar HSS, Dhangadhi
17	Raj Kamal Bhandari	Dhangadhi HSS, Dhangadhi
18	Prabhuram Chaudhari	Dipendra HSS, Hasuliya

All these teachers were participants in the TPD training of 2069/70 at ETC 'B' , Dhangadhi, Kailali and all of them filled the demand proposal form before coming to this training. Some of these teachers have put their demand on the same topic i.e. nth term of a sequence.

Planning and Implementation
i) Study the structure/pattern of the various sequences
ii) Limitations of the formula for nth term of AS and GS
iii) Consult reference materials from various books and websites
iv) Organize the matter and use it in the training class.

i) Study the structure/pattern of the various sequences Case I :

Let us have a sequence 1, 3, 5, 7, …, …, …

Here d = 2, so nth term will be multiple of 2 and to find first term = 1 from 2n (for n = 1), we should subtract 1.

Thus nth term will be 2n-1.

We can also check as follows :

t1 = 1 = 2.1-1 , t2 = 3= 2.2-1

t3 = 5 = 2.3 -1 , t4 = 7 = 2.4 -1

…. … … , tn = 2.n-1

Case II : Let us have another sequence

1, 2, 4, 8, 16, … , …,

t1 = 1.21-1= 1.20 = 1.1 = 1,

t2= 1.22-1= 1.21= 1.2= 2

similarly tn = 1.2n-1 = 2n-1

Direct formulae for arithmetic sequences is tn = a + (n-1)d and for geometric sequcences is tn = arn-1, but the important thing to remember is that we use the induction method rather than the direct formula. However, the formula helps us to think of the certain pattern and it saves our time and effort and we can easily guide the students to go in a appropriate way.

ii) Limitations of the formula for nth term of AS and GS

Let us have another type of sequence

10, 12, 16, 22, 30, …, …

Note that this sequence is neither in AS nor in GS and the formula which we used above do not work here. This is the limitation of the formulae. At this stage again either we have to use induction method or search for a new formula.

iii) Findings from reference materials, various books and websites

Finally I made up mind to go under the shelter of internet. I got the following formulae, tips and conclusions :

1. Common difference formula/when the first differences are constant (i.e. the sequence is arithmetic)

Imagine the sequence 2, 4, 6, 8, 10, …

The formula used here is a + (n-1)d

2. For geometric sequence tn = arn-1

3. Changing difference formula

Here,specially When the II differences are constant,

tn = a +(n – 1)d + 0.5(n -1)(n -2) C

Where d = the first difference =t2- t1,

C = c. d. of the sequence of first differences

Here d = 5 -3 =2, c = 2

Nth term = 3 + (n – 1)x2 + 0.5(n – 1)(n – 2)x2

= 3 + 2n – 2 + n2 - 3n + 2 = n2 - n + 3

Implementation

To use this information in the training classroom, I porepared some worksheets, wrote t he related formulae on it and divided the students in 6 groups of 3 participants in each.The group division was done on the following basis :

Group serial no. (above table of name list)

A (1, 3, 5)

B (7, 9, 11)

C (13, 15, 17)

D (2, 4, 6)

E (8, 10, 12)

F (14, 16, 18)

The worksheet was same for each group and was as follows :

Find the nth term of the following sequences :

i) 8, 3, -2, -7, -12, …, …,

ii) 3, 6, 12, 24, 48, …, …

iii) 4, 9, 16, 25, 36, …, …

iv) 5, 10, 17, 26, 37, 50, …, …

v) 6, 11, 18, 27, 38, 51, …, …

vi) 3, 8, 15, 24, 35, 48, …, …

vii) -4, 9, -16, 25, -36, 49, …, …

viii) 4, -9, 16, -25, 36, …, …

ix) 2/4, 5/9, 8/16, 11/25, …, …

x) -5, 7/2, -9/3, 11/4, -13/5, …, …

xi) 1, 1/2 , 1/3, 1/4, 1/5, …, …

xii) 1/2, 2/3, 3/4, 4/5, …, …

xiii) -3, -1, -1/3, 0, 1/5, …, …

xiv) -1, 2, -3, 4, -5, …, …

xv) 3, -2, 5/3, -3/2, 7/5, …, …

xvi) 1, 1/2, 3, 1/4, 5, …, …

xvii) 1/2, -2/6, 3/18, -4/54, 5/162, …, …

xviii) 5/6, 7/12, 9/24, 11/ 48, 13/96, …, …

xix) 1, 5, 14, 30, …, …

xx) 1, 9, 36, 100, …, …

Conclusion

After the performing the group work, an individual evaluation was also conducted and each group achieved the competency in finding the general term of the given sequence. They also made the following conclusions :

1. Check if the numbers are square numbers like 4, 9, 16, 25, 36, …, write them as 22, 32, 42, 52, … Note that bases are in A.S., where d = 1 and exponent is a constant, so bn= 2+ (n-1)x1= n+1

and so tn = (n+1)2

2) If the second differences are equal then the nth term begins with n2 i.e. for 1, 4, 9, 16, …,

tn = n2

3) If third differences are equal then nth term begins with n3 i.e. for 1, 8, 27, 64, …,

tn = n3

4) If the terms of sequence are numbers next to perfect squares i.e.

5, 10, 17, 26, 37, …

= 22+1, 32+1, 42+1, …

Now find nth term as in above example and add 1.

Thus tn = (n+1)2 +1

5) Similarly for 6, 11, 18, 27, 38, …

= 22+2, 32+2, 42+2, …

tn = (n+1)2 +2

6) For the sequence 3, 8, 15, 24, … =22-1, 32-1, 42-1, …

tn = (n+1)2 -1

7) Check if the terms of the sequence are fractional and it is not an arithmetic or geometric sequence, then calculate the nth term of the numerator and denominator separately. i.e. for 2/4, 5/9, 11/25, …,

here tn = (3n-1)/(n+1)2

8)Note that there is no easy way of working out the nth term of a sequence other than to try different possibilities.

Unsolved topic that will be studied in the next cycle

Although the trainee teachers got the skill to find the general term for the sequence which has first and second differences constant, but the above explanation doesn't solve the problem if the third or fourth or further differences are equal/constant.

TPD Training Participants 2069/70, Mathematics(Secondary Level)

S.N.	Name of Teachers	Name of School
1	Puskar Raj Bhatta	Saraswati HSS, Geta
2	Govind Prasad Bhatta	Sharada HSS, Baiyabehedi
3	Bikalpa Chaudhari	Laxmi HSS, Udasipur
4	Asharam Chaudhari	Mahendra Shishu HSS, Dharmapur
5	Rajendra Pokharel	Khadgasmriti HSS, Tikapur
6	Jang Bahadur Bam	Bhanu ma.vi. Pathari
7	Harendraraj Awasthi	Janata Ra.ma.vi. Kailali gaun
8	Gyanendraraj Joshi	Trinagar HSS, Dhangadhi
9	Uddhav Dev Bhatt	Jankalyan HSS, Jkugeda
10	Janak Prasad Bhatt	Phulbari HSS, Phulbari
11	Karna Bahadur Kadayat	Gwasi HSS, Malakheti
12	Gorakh Nath	Shaileswori HSS, Badeha
13	Radha Bisht	Saraswati HSS, Pratappur
14	Raj Bahadur Kadayat	Siddsha Baba Ma.vi. Geti
15	Ganesh Datt Joshi	Panchoday HSS, Dhangadhi
16	Devendra Jha	Trinagar HSS, Dhangadhi
17	Raj Kamal Bhandari	Dhangadhi HSS, Dhangadhi
18	Prabhuram Chaudhari	Dipendra HSS, Hasuliya

Multilingual education(MLE) training, conducted at etc, kailali from 8th baisakh 2070 is over today. There 26 participants of different primary schools of kailali district. There were two trainers to facilitate the class : Mr. Yadav Prasad Joshi and me (Krishna Singh Pela)

गणित शिक्षण/प्रशिक्षणमा खेल तथा पजलहरुको प्रयोग

गणित शिक्षणलाई रुचिपूर्ण र प्रभावकारी बनाउनका लागि कक्षाकोठामा गणितीय खेल तथा पजलहरुको प्रयोग अत्यन्त जरुरी देखिन्छ । एकोहोरो व्याख्यान विधि वा समस्या समाधान विधिले विद्यार्थीहरुको ध्यान सिकाइबाट विचलित हुन जान्छ । गणित विषयमा अन्य रोचक प्रसंगहरु वा कथा कहानीहरुको अभाव हुनु र सधैं समस्या समस्या समाधान गर्ने कार्यमात्र भइराख्नुले यस विषयबाट सिकारुको ध्यान पलायन हुनु स्वभाविकै हो । दैनिक जीवनमा गणितको उपयोगिताको अनुभूति हुनु जति आवश्यक छ, गणितलाई रोचक बनाउने विभिन्न उपायहरु खोज्नु तथा तिनलाई उपयोग गर्नु पनि त्यतिकै अपरिहार्य छ । यसका लागि छुट्टै समय माग गर्नुभन्दा पनि गणितकै सहक्रियाकलापको रुपमा बीचबीचमा गणितीय विषयवस्तुलाई खेल वा पजलका माध्यमबाट प्रस्तुत गर्न सकियो भने कक्षाकोठाको वातावरण सक्रिय, जीवंत, रोचक र अर्थपूर्ण बनाउन सकिन्छ ।

गणितीय पजलको अर्थ

गणितीय पजल भनेको गणितीय समस्या नै हो । जुनसुकै समस्या जसले मस्तिष्कमन्थन गराउँछ, त्यो पजल हो भने यस्तो समस्या जसको समाधानको प्रक्रिया गणितीय सिद्धांतमा आधारित छ , त्यो गणितीय पजल हो । यस्तो पजल कुनै सूत्र वा गणितीय क्रियामा आधारित पनि हुनसक्छ र केवल Tricks मा मात्रै आधारित पनि हुनसक्छ जसलाई Inspection Method या Induction Method या Trial and Error Method बाट समाधान गर्न सकिन्छ ।

गणितीय पजलको महत्व

1. यसले तर्कशक्तिको विकास गराउँछ
2. चिन्तनको क्षमतालाई सूक्ष्म र तीव्र बनउँछ
3. गणितीय ज्ञानलाई फराकिलो बनाउँछ
4. आत्मविश्वास बढाउँछ
5. गणित कक्षालाई मनोरञ्जनपूर्ण बनाउँछ
6. पजलसँग सम्बन्धित गणितीय धारणाको विकास गराउँछ
गणितीय खेल
कुनै क्रियाकलापलाई खेल तबमात्र भनिन्छ जब यसमा निम्नलिखित विशेषताहरु हुन्छन् ः
1. कम्तीमा दुईओटा व्यक्ति वा दुईओटा समूहको सहभागिता भएको
2. सहभागी व्यक्ति वा समूहबीच प्रतिस्पर्धा भएको
3. निश्चित नियमहरु निर्धारण गरिएको र त्यही नियमानुसार खेलिएको
4. अन्त्यमा एउटा व्यक्ति वा समूह विजयी हुने कुरा निश्चित भएको
5. विजयी व्यक्ति वा समूहका लागि पुरस्कारको व्यवस्था गरिएको गणितीय खेलको प्रयोग गर्दा ध्यान दिनुपर्ने कुराहरु
1. निर्धारित सबै जनाको सहभागिता हुनुपर्छ
2. गणितीय खेललाई गणित विषयको सहक्रियाकलापको रुपमा सञ्चालन गर्नुपर्छ
3. दिक्क लाग्ने खालको हुनु हुँदैन
4. समय धेरै लाग्ने हुनु हुँदैन
5. ीनर्धारित समयमा पूरा हुने खालको हुनुपर्छ
6. सबै विद्याथीहरुले खेलको नियमलाई अनिवार्य रुपमा पालना गरेको हुनुपर्छ
7. एपयुक्त समयमा मात्र सञ्चालन गर्नुपर्छ अर्थात् दिनहुँ खेल खेलाउनु जरुरी छैन, विषयवस्तु हेरेर आवश्यक भएमा र विद्यार्थीहरुको रुचि भएमा मात्र खेलाउनु पर्छ ।
गणितीय खेलको महत्व
1. शिक्षणलाई रुचिकर र आकर्षक बनाउँछ
2. विषयवस्तुको धारण सजिलै दिन सकिन्छ
3. मनोरञ्जनपूर्ण सिकाइ हुन्छ
4. सिर्जनशीलतामा वृद्धि गराउँछ
5. थप सिकाइका लागि उत्प्रेरित गराउँछ
6. गणितप्रति अभिरुचि जगाउँछ
7. सहनशील र धैर्यवान बनाउँछ

Project Work and Assignment for Online Offline Training

Online Offline Training conducted at ETC 'B', Kailali, from 2069/12/15 to 2069/12/21. Out of many topics discusssd during this training, project works related to different themes were also under discussion. The samples for reference are described here in brief. 

1.     TPD self learning material 2068

Project work

1.Design a session for Large Classroom management, use it in TPD training class and write a report including the participants' feedback.

2. Make a collection of mathematical games and puzzles together with the TPD participants so that they will be able to construct and use this in their classroom to motivate the students towards mathematics. Also write a report with the feedback of the participants.

Assignment

1.Clarify the concept of Active Paricipatory Approach and how it can be used in TPD training ?

2. Mention the evaluation tools to evaluate the training conducted in your ETC. Describe how these tools can be used and prepare any one of these tools.

2.   Generic self learning material 2068

Project work

1.    Make a Rubrics to evaluate the 5 days' TPD training and use it in the TPD training classroom so that the trainee teachers would be able to construct the Rubrics for evaluating the students' performance. Also make a report together with the feedback and output of this work

2.    List the probable topics i.e. historical places, temples, rivers, lakes or from any other agriculture sector to construct the local curriculum. Now collect the information about the related topics and make a framework of local curriculum with the help of primary curruculum guide. Use it in TPD training and write a report including the participants' feedback that how it is useful to them.

Assignment

1.    Clarify the concept of child friendly school. What are the fundamental aspects to make a school child friendly and write down the challenges that are to be faced during this process.

2.    What types of social diversities are found in the schools of the west terai region ? How can we call these diversities in the classroom ?

3.    ICT self learning material 2068

Project work

1.     Design a session for TPD training and prepare it in microsoft powerpoint. Use this session and write a report including the feedback of the participants.

2.     Download the reference material from the related websites for the concept of muultiplication of negative numbers. Write a report how this material helped you as a trainer in the TPD training class and also write down the mehods how you found that material ?

Assignment

1.    What do you mean by search engines ? Mention the list of important and usually used search engines. Write the applications of search engines with illustration.

2.    Define email and internet. How can you create an email account. Create an email account and write the procedures to create it. Also describe the use of email and internet and how is it useful in teachers' professional development ?

4.    HT LCBT self learning material 2068

Project work

1.     Construct a tool for Time management and use it in Head Teachers' training. Share with them that how can we manage our time and also write a report submitting their feedback.

2.     Develop the forms for different accounting system of the schools. Distribute these forms to participants and visit the nearest school of the ETC in order to fill those forms with related informations, i.e. how the school has kept its accounts. Also make a report that what benifit is obtained from this task ?

Assignment

1.     What do you mean by SIP ? Write it's importance. How the quality of a school can be improved ?

2.     What are the main theories of leadership ? Differentiate between Gregor's X theory and Y theory. Compare between transactional and transformational leadership.