IJCRR

International Journal of Current Research and Review
ISSN: 2231-2196 (Print)ISSN: 0975-5241 (Online)
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IJCRR - 9(10) Special Issue, May, 2017

Pages: 68-70

Power Sums Through Mathematical Induction

Author: R. Malathi, C. Tamizharasi

Category: General Sciences

Abstract:

We observe that Faulhaber’s theorem on sums of odd powers holds an random arithmetic progression, the odd power sums of any arithmetic progression: a+b, a+2b,…, a+nb is a polynomial in n a + n(n+1)b/2. This assertion can be presumed from the original Faulhaber’s theorem. We use the Bernoulli polynomials for the alternative formula. By using the central factorial numbers as in the approach of knuth, we can derive the formulas for r – fold sums of powers without restoring the notion of r – reflective functions. We can also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials. In its simplest case, a power sum is a sum of the form Sn(l) = 1n + 2n + . . . . . +(l-1)n
Their sums have interesting combination of a number theoretical importance, and were already known by Bernoulli’s, but we shall see that even today there are many things about them to discover.
• Basics of power sums
• Generalizations of power sums
• Alternating power sums
In this paper, we are going to discuss about the negative power sum.

Keywords: Infinite sum, Partial sums, Euler polynomial

Citation:

R. Malathi, C. Tamizharasi. Power Sums Through Mathematical Induction International Journal of Current Research and Review. 9(10) Special Issue, May, 68-70

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Dr. Pramod Kumar Manjhi joined Editor-in-Chief since July 2021 onwards

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Awards, Research and Publication incentive Schemes by IJCRR

Best Article Award:

One article from every issue is selected for the ‘Best Article Award’. Authors of selected ‘Best Article’ are rewarded with a certificate. IJCRR Editorial Board members select one ‘Best Article’ from the published issue based on originality, novelty, social usefulness of the work. The corresponding author of selected ‘Best Article Award’ is communicated and information of award is displayed on IJCRR’s website. Drop a mail to editor@ijcrr.com for more details.

Women Researcher Award:

This award is instituted to encourage women researchers to publish her work in IJCRR. Women researcher, who intends to publish her research work in IJCRR as the first author is eligible to apply for this award. Editorial Board members decide on the selection of women researchers based on the originality, novelty, and social contribution of the research work. The corresponding author of the selected manuscript is communicated and information is displayed on IJCRR’s website. Under this award selected women, the author is eligible for publication incentives. Drop a mail to editor@ijcrr.com for more details.

Emerging Researcher Award:

‘Emerging Researcher Award’ is instituted to encourage student researchers to publish their work in IJCRR. Student researchers, who intend to publish their research or review work in IJCRR as the first author are eligible to apply for this award. Editorial Board members decide on the selection of student researchers for the said award based on originality, novelty, and social applicability of the research work. Under this award selected student researcher is eligible for publication incentives. Drop a mail to editor@ijcrr.com for more details.

Best Article Award

A study by Dorothy Ebere Adimora et al. entitled \"Remediation for Effects of Domestic Violence on Psychological well-being, Depression and Suicide among Women During COVID-19 Pandemic: A Cross-cultural Study of Nigeria and Spain\" is awarded Best Article of Vol 14 issue 23
A study by Muhas C. et al. entitled \"Study on Knowledge & Awareness About Pharmacovigilance Among Pharmacists in South India\" is awarded Best article for Vol 14 issue 22
A study by Saurabh Suvidha entitled \"A Case of Mucoid Degeneration of Uterine Fibroid with Hydrosalphinx and Ovarian Cyst\" is awarded Best article of Vol 14 issue 21
A study by Alice Alice entitled \"Strengthening of Human Milk Banking across South Asian Countries: A Next Step Forward\" is awarded Best article of Vol 14 issue 20
A study by Sathyanarayanan AR et al. entitled \"The on-task Attention of Individuals with Autism Spectrum Disorder-An Eye Tracker Study Using Auticare\" is awarded Best article of Vol 14 issue 19
A study by Gupta P. et al. entitled \"A Short Review on \"A Novel Approach in Fast Dissolving Film & their Evaluation Studies\" is awarded Best Article of Vol 14 issue 18.
A study by Shafaque M. et al. entitled \"A Case-Control Study Performed in Karachi on Inflammatory Markers by Ciprofloxacin and CoAmoxicillin in Patients with Chronic Suppurative Otitis Media\" is awarded Best Article of Vol 14 issue 17
A study by Ali Nawaz et al. entitled \"A Comparative Study of Tubeless versus Standard Percutaneous Nephrolithotomy (PCNL) \? A Randomized Controlled Study\" is awarded Best Article for Vol 14 issue 16.
A study by Singh R. et al. entitled \"A Prospective Study to Find the Association of Astigmatism in Patients of Vernal Keratoconjunctivitis (VKC) in a Tertiary Health Care Centre in India (Vindhya Region MP)\" is awarded Best Article for Vol 14 issue 15
A Study by Humaira Tahir et al. entitled "Comparison of First Analgesic Demand after Major Surgeries of Obstetrics and Gynecology between Pre-Emptive Versus Intra-Operative Groups by Using Intravenous Paracetamol: A Cross-Sectional Study" is awarded Best Article for Vol 14 issue 14
A Study by Monica K. entitled "Risk Predictors for Lymphoma Development in Sjogren Syndrome - A Systematic Review" is awarded Best Article for Vol 14 issue 13
A Study by Mokhtar M Sh et al. entitled "Prevalence of Hospital Mortality of Critically Ill Elderly Patients" is awarded Best Article for Vol 14 issue 12
A Study by Vidya S. Bhat et al. entitled "Effect of an Indigenous Cleanser on the Microbial Biofilm on Acrylic Denture Base - A Pilot Study" is awarded Best Article for Vol 14 issue 11
A Study by Pandya S. et al. entitled "Acute and 28-Day Repeated Dose Subacute Toxicological Evaluation of Coroprotect Tablet in Rodents" is awarded Best Article for Vol 14 issue 10
A Study by Muhammad Zaki et al. entitled "Effect of Hemoglobin Level on the Severity of Acute Bronchiolitis in Children: A Case-Control Study" is awarded Best Article for Vol 14 issue 09
A Study by Vinita S & Ayushi S entitled "Role of Colour Doppler and Transvaginal Sonography for diagnosis of endometrial pathology in women presenting with Abnormal Uterine Bleeding" is awarded Best Article for Vol 14 issue 08
A Study by Prabhu A et al. entitled "Awareness of Common Eye Conditions among the ASHA (Accredited Social Health Activist) Workers in the Rural Communities of Udupi District- A Pilot Study" is awarded Best Article for Vol 14 issue 07
A Study by Divya MP et al. entitled "Non-Echoplanar Diffusion-Weighted Imaging and 3D Fiesta Magnetic Resonance Imaging Sequences with High Resolution Computed Tomography Temporal Bone in Assessment and Predicting the Outcome of Chronic Suppurative Otitis Media with Cholesteatoma" is awarded Best Article for Vol 14 issue 06
A Study by Zahoor Illahi Soomro et al. entitled "Functional Outcomes of Fracture Distal Radius after Fixation with Two Different Plates: A Retrospective Comparative Study" is awarded Best Article for Vol 14 issue 05
A Study by Ajai KG & Athira KN entitled "Patients’ Gratification Towards Service Delivery Among Government Hospitals with Particular Orientation Towards Primary Health Centres" is awarded Best Article for Vol 14 issue 04
A Study by Mbungu Mulaila AP et al. entitled "Ovarian Pregnancy in Kindu City, D.R. Congo - A Case Report" is awarded Best Article for Vol 14 issue 03
A Study by Maryam MJ et al. entitled "Evaluation Serum Chemerin and Visfatin Levels with Rheumatoid Arthritis: Possible Diagnostic Biomarkers" is awarded Best Article for Vol 14 issue 02
A Study by Shanthan KR et al. entitled "Comparison of Ultrasound Guided Versus Nerve Stimulator Guided Technique of Supraclavicular Brachial Plexus Block in Patients Undergoing Upper Limb Surgeries" is awarded Best Article for Vol 14 issue 01
A Study by Amol Sanap et al. entitled "The Outcome of Coxofemoral Bypass Using Cemented Bipolar Hemiarthroplasty in the Treatment of Unstable Intertrochanteric Fracture of Femur in a Rural Setup" is awarded Best Article Award of Vol 13 issue 24
A Study by Manoj KP et al. entitled "A Randomized Comparative Clinical Trial to Know the Efficacy of Ultrasound-Guided Transversus Abdominis Plane Block Against Multimodal Analgesia for Postoperative Analgesia Following Caesarean Section" is awarded Best Article Award of Vol 13 issue 23
A Study by Karimova II et al. entitled "Changes in the Activity of Intestinal Carbohydrases in Alloxan-Induced Diabetic Rats and Their Correction with Prenalon" is awarded Best Article of Vol 13 issue 22
A Study by Ashish B Roge et al. entitled "Development, Validation of RP-HPLC Method and GC MS Analysis of Desloratadine HCL and It’s Degradation Products" is awarded Best Article of Vol 13 issue 21
A Study by Isha Gaurav et al. entitled "Association of ABO Blood Group with Oral Cancer and Precancer – A Case-control Study" is awarded Best Article for Vol 13 issue 20
A Study by Amr Y. Zakaria et al. entitled "Single Nucleotide Polymorphisms of ATP-Binding Cassette Gene(ABCC3 rs4793665) affect High Dose Methotrexate-Induced Nephrotoxicity in Children with Osteosarcoma" is awarded Best Article for Vol 13 issue 19
A Study by Kholis Ernawati et al. entitled "The Utilization of Mobile-Based Information Technology in the Management of Dengue Fever in the Community Year 2019-2020: Systematic Review" is awarded Best Article for Vol 13 issue 18
A Study by Bhat Asifa et al. entitled "Efficacy of Modified Carbapenem Inactivation Method for Carbapenemase Detection and Comparative Evaluation with Polymerase Chain Reaction for the Identification of Carbapenemase Producing Klebsiella pneumonia Isolates" is awarded Best Article for Vol 13 issue 17
A Study by Gupta R. et al. entitled "A Clinical Study of Paediatric Tracheostomy: Our Experience in a Tertiary Care Hospital in North India" is awarded Best Article for Vol 13 issue 16
A Study by Chandran Anand et al. entitled "A Prospective Study on Assessment of Quality of Life of Patients Receiving Sorafenib for Hepatocellular Carcinoma" is awarded Best article for Vol 13 issue 15
A Study by Rosa PS et al. entitled "Emotional State Due to the Covid – 19 Pandemic in People Residing in a Vulnerable Area in North Lima" is awarded Best Article for Vol 13 issue 14
A Study by Suvarna Sunder J et al. entitled "Endodontic Revascularization of Necrotic Permanent Anterior Tooth with Platelet Rich Fibrin, Platelet Rich Plasma, and Blood Clot - A Comparative Study" is awarded Best Article for Vol 13 issue 13
A Study by Mona Isam Eldin Osman et al. entitled "Psychological Impact and Risk Factors of Sexual Abuse on Sudanese Children in Khartoum State" is awarded Best Article for Vol 13 issue 12
A Study by Khaw Ming Sheng & Sathiapriya Ramiah entitled "Web Based Suicide Prevention Application for Patients Suffering from Depression" is awarded Best Article for Vol 13 issue 11
A Study by Purushottam S. G. et al. entitled "Development of Fenofibrate Solid Dispersions for the Plausible Aqueous Solubility Augmentation of this BCS Class-II Drug" is awarded Best article for Vol 13 issue 10
A Study by Kumar S. et al. entitled "A Study on Clinical Spectrum, Laboratory Profile, Complications and Outcome of Pediatric Scrub Typhus Patients Admitted to an Intensive Care Unit from a Tertiary Care Hospital from Eastern India" is awarded Best Article for Vol 13 issue 09
A Study by Mardhiah Kamaruddin et al. entitled "The Pattern of Creatinine Clearance in Gestational and Chronic Hypertension Women from the Third Trimester to 12 Weeks Postpartum" is awarded Best Article for Vol 13 issue 08
A Study by Sarmila G. B. et al. entitled "Study to Compare the Efficacy of Orally Administered Melatonin and Clonidine for Attenuation of Hemodynamic Response During Laryngoscopy and Endotracheal Intubation in Gastrointestinal Surgeries" is awarded Best Article for Vol 13 issue 07
A Study by M. Muthu Uma Maheswari et al. entitled "A Study on C-reactive Protein and Liver Function Tests in Laboratory RT-PCR Positive Covid-19 Patients in a Tertiary Care Centre – A Retrospective Study" is awarded Best Article of Vol 13 issue 06 Special issue Modern approaches for diagnosis of COVID-19 and current status of awareness
A Study by Gainneos PD et al. entitled "A Comparative Evaluation of the Levels of Salivary IgA in HIV Affected Children and the Children of the General Population within the Age Group of 9 – 12 Years – A Cross-Sectional Study" is awarded Best Article of Vol 13 issue 05 Special issue on Recent Advances in Dentistry for better Oral Health
A Study by Alkhansa Mahmoud et al. entitled "mRNA Expression of Somatostatin Receptors (1-5) in MCF7 and MDA-MB231 Breast Cancer Cells" is awarded Best Article of Vol 13 issue 06
A Study by Chen YY and Ghazali SRB entitled "Lifetime Trauma, posttraumatic stress disorder Symptoms and Early Adolescence Risk Factors for Poor Physical Health Outcome Among Malaysian Adolescents" is awarded Best Article of Vol 13 issue 04 Special issue on Current Updates in Plant Biology to Medicine to Healthcare Awareness in Malaysia
A Study by Kumari PM et al. entitled "Study to Evaluate the Adverse Drug Reactions in a Tertiary Care Teaching Hospital in Tamilnadu - A Cross-Sectional Study" is awarded Best Article for Vol 13 issue 05
A Study by Anu et al. entitled "Effectiveness of Cytological Scoring Systems for Evaluation of Breast Lesion Cytology with its Histopathological Correlation" is awarded Best Article of Vol 13 issue 04
A Study by Sharipov R. Kh. et al. entitled "Interaction of Correction of Lipid Peroxidation Disorders with Oxibral" is awarded Best Article of Vol 13 issue 03
A Study by Tarek Elwakil et al. entitled "Led Light Photobiomodulation Effect on Wound Healing Combined with Phenytoin in Mice Model" is awarded Best Article of Vol 13 issue 02
A Study by Mohita Ray et al. entitled "Accuracy of Intra-Operative Frozen Section Consultation of Gastrointestinal Biopsy Samples in Correlation with the Final Histopathological Diagnosis" is awarded Best Article for Vol 13 issue 01
A Study by Badritdinova MN et al. entitled "Peculiarities of a Pain in Patients with Ischemic Heart Disease in the Presence of Individual Combines of the Metabolic Syndrome" is awarded Best Article for Vol 12 issue 24
A Study by Sindhu Priya E S et al. entitled "Neuroprotective activity of Pyrazolone Derivatives Against Paraquat-induced Oxidative Stress and Locomotor Impairment in Drosophila melanogaster" is awarded Best Article for Vol 12 issue 23
A Study by Habiba Suhail et al. entitled "Effect of Majoon Murmakki in Dysmenorrhoea (Usre Tams): A Standard Controlled Clinical Study" is awarded Best Article for Vol 12 issue 22
A Study by Ghaffar UB et al. entitled "Correlation between Height and Foot Length in Saudi Population in Majmaah, Saudi Arabia" is awarded Best Article for Vol 12 issue 21
A Study by Siti Sarah Binti Maidin entitled "Sleep Well: Mobile Application to Address Sleeping Problems" is awarded Best Article for Vol 12 issue 20
A Study by Avijit Singh"Comparison of Post Operative Clinical Outcomes Between “Made in India” TTK Chitra Mechanical Heart Valve Versus St Jude Mechanical Heart Valve in Valve Replacement Surgery" is awarded Best Article for Vol 12 issue 19
A Study by Sonali Banerjee and Mary Mathews N. entitled "Exploring Quality of Life and Perceived Experiences Among Couples Undergoing Fertility Treatment in Western India: A Mixed Methodology" is awarded Best Article for Vol 12 issue 18
A Study by Jabbar Desai et al. entitled "Prevalence of Obstructive Airway Disease in Patients with Ischemic Heart Disease and Hypertension" is awarded Best Article for Vol 12 issue 17
A Study by Juna Byun et al. entitled "Study on Difference in Coronavirus-19 Related Anxiety between Face-to-face and Non-face-to-face Classes among University Students in South Korea" is awarded Best Article for Vol 12 issue 16
A Study by Sudha Ramachandra & Vinay Chavan entitled "Enhanced-Hybrid-Age Layered Population Structure (E-Hybrid-ALPS): A Genetic Algorithm with Adaptive Crossover for Molecular Docking Studies of Drug Discovery Process" is awarded Best article for Vol 12 issue 15
A Study by Varsha M. Shindhe et al. entitled "A Study on Effect of Smokeless Tobacco on Pulmonary Function Tests in Class IV Workers of USM-KLE (Universiti Sains Malaysia-Karnataka Lingayat Education Society) International Medical Programme, Belagavi" is awarded Best article of Vol 12 issue 14, July 2020
A study by Amruta Choudhary et al. entitled "Family Planning Knowledge, Attitude and Practice Among Women of Reproductive Age from Rural Area of Central India" is awarded Best Article for special issue "Modern Therapeutics Applications"
A study by Raunak Das entitled "Study of Cardiovascular Dysfunctions in Interstitial Lung Diseas epatients by Correlating the Levels of Serum NT PRO BNP and Microalbuminuria (Biomarkers of Cardiovascular Dysfunction) with Echocardiographic, Bronchoscopic and HighResolution Computed Tomography Findings of These ILD Patients" is awarded Best Article of Vol 12 issue 13
A Study by Kannamani Ramasamy et al. entitled "COVID-19 Situation at Chennai City – Forecasting for the Better Pandemic Management" is awarded best article for  Vol 12 issue 12
A Study by Muhammet Lutfi SELCUK and Fatma entitled "Distinction of Gray and White Matter for Some Histological Staining Methods in New Zealand Rabbit's Brain" is awarded best article for  Vol 12 issue 11
A Study by Anamul Haq et al. entitled "Etiology of Abnormal Uterine Bleeding in Adolescents – Emphasis Upon Polycystic Ovarian Syndrome" is awarded best article for  Vol 12 issue 10
A Study by entitled "Estimation of Reference Interval of Serum Progesterone During Three Trimesters of Normal Pregnancy in a Tertiary Care Hospital of Kolkata" is awarded best article for  Vol 12 issue 09
A Study by Ilona Gracie De Souza & Pavan Kumar G. entitled "Effect of Releasing Myofascial Chain in Patients with Patellofemoral Pain Syndrome - A Randomized Clinical Trial" is awarded best article for  Vol 12 issue 08
A Study by Virendra Atam et. al. entitled "Clinical Profile and Short - Term Mortality Predictors in Acute Stroke with Emphasis on Stress Hyperglycemia and THRIVE Score : An Observational Study" is awarded best article for  Vol 12 issue 07
A Study by K. Krupashree et. al. entitled "Protective Effects of Picrorhizakurroa Against Fumonisin B1 Induced Hepatotoxicity in Mice" is awarded best article for issue Vol 10 issue 20
A study by Mithun K.P. et al "Larvicidal Activity of Crude Solanum Nigrum Leaf and Berries Extract Against Dengue Vector-Aedesaegypti" is awarded Best Article for Vol 10 issue 14 of IJCRR
A study by Asha Menon "Women in Child Care and Early Education: Truly Nontraditional Work" is awarded Best Article for Vol 10 issue 13
A study by Deep J. M. "Prevalence of Molar-Incisor Hypomineralization in 7-13 Years Old Children of Biratnagar, Nepal: A Cross Sectional Study" is awarded Best Article for Vol 10 issue 11 of IJCRR
A review by Chitra et al to analyse relation between Obesity and Type 2 diabetes is awarded 'Best Article' for Vol 10 issue 10 by IJCRR.
A study by Karanpreet et al "Pregnancy Induced Hypertension: A Study on Its Multisystem Involvement" is given Best Paper Award for Vol 10 issue 09

List of Awardees

A Study by Ese Anibor et al. "Evaluation of Temporomandibular Joint Disorders Among Delta State University Students in Abraka, Nigeria" from Vol 13 issue 16 received Emerging Researcher Award

A Study by Alkhansa Mahmoud et al. entitled "mRNA Expression of Somatostatin Receptors (1-5) in MCF7 and MDA-MB231 Breast Cancer Cells" from Vol 13 issue 06 received Emerging Researcher Award

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Disclaimer: International Journal of Current Research and Review (IJCRR) provides platform for researchers to publish and discuss their original research and review work. IJCRR can not be held responsible for views, opinions and written statements of researchers published in this journal.

International Journal of Current Research and Review (IJCRR) provides platform for researchers to publish and discuss their original research and review work. IJCRR can not be held responsible for views, opinions and written statements of researchers published in this journal