﻿ A P-Series Formula Applied Mathematics

p-ISSN: 2163-1409    e-ISSN: 2163-1425

2021;  11(2): 21-22

doi:10.5923/j.am.20211102.02

Received: Jun. 14, 2021; Accepted: Jul. 11, 2021; Published: Jul. 15, 2021 ### A P-Series Formula

Ameha Tefera Tessema

Commercial Bank of Ethiopia, Addis Ababa, Ethiopia

Correspondence to: Ameha Tefera Tessema, Commercial Bank of Ethiopia, Addis Ababa, Ethiopia.
 Email:  Abstract

Let such that then   Where p=2t+1 for all p odd Natural numbers(N), p=2t+2 for all p even natural numbers(N).

Keywords: Power series, Formula

Cite this paper: Ameha Tefera Tessema, A P-Series Formula, Applied Mathematics, Vol. 11 No. 2, 2021, pp. 21-22. doi: 10.5923/j.am.20211102.02.

1. Introduction

### 1. Introduction

In the early time fermat, a French number theorist, had stated that the equation has no solution in integers, if n>2 . The problem to find the exact value of z is enhanced as the number of terms increases rather than power two. Let us have n terms each of which has a common power p in the series then the series is called a p-series which is applied by the following theorem for p>2 which is also an answer for the claim of fermat.
Theorem 1. Let such that then Where p=2t+1 for all p odd Natural numbers(N), p=2t+2 for all p even natural numbers.
Proof. Let such that if where then multiplying up to n times we get , therefor we can write this expression to p power as If the binomial theorem for expansion .
Can be written as Where ,
Putting and dividing both sides by we get (1)
The other way of writing is which can be written to p power as If binomial theorem for expansion can be written as Where ,
Putting in and dividing both sides by we get (2)
Therefor, the summation of EQ(1) and EQ(2) can be put as follow Where p=2t+1 for all p odd natural numbers, whereas p=2t+2 for all p even natural numbers
Let put and if be divided by such that (Real numbers) then we get the following result This result led us to a conclusion that every couple terms which have common power can undergo this formula.

### References

  Gerhard Frey, The way to the proof of Fermat’s Last Theorem, Annales de la faculte des sciences de Toulouse, Vol. XVIII, 2009, pp.5-23.