My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum

LinkBack Thread Tools Display Modes
May 1st, 2014, 01:03 AM   #1
Senior Member
Joined: Sep 2013
From: Earth

Posts: 827
Thanks: 36

Arithmetic Progression and Geometric Progression

How to proof those formula in these two progressions?
jiasyuen is offline  
May 1st, 2014, 04:40 AM   #2
Senior Member
Olinguito's Avatar
Joined: Apr 2014
From: Greater London, England, UK

Posts: 320
Thanks: 156

Math Focus: Abstract algebra
(1) Arithmetic progression with first term $a$ and common difference $d$.

The $n$th term is $T_n=a+(n-1)d$. Thus
$\displaystyle =\ \sum_{k=1}^n\,[a+(k-1)d]$

$\displaystyle =\ \sum_{k=1}^n\,a \, + \, d\sum_{k=1}^n\,(k-1)$

$\displaystyle =\ na+d\frac{(n-1)n}2$

$\displaystyle =\ \frac n2\left[2a+(n-1)d\right]$

(2) Geometric progression with first term $a$ and common difference $r$.

Case 1: $r=1$
In this case the progression is a constant sequence consisting of $a$ in every term so the sum to $n$th term is $na$.

Case 2: $r\ne1$
The $n$th term is $T_n=ar^{n-1}$. Thus $(r-1)T_n=a(r^n-r^{n-1})$ and
$\displaystyle =\ a\sum_{k=1}^n\,(r^k-r^{k-1})$

$\displaystyle =\ a(r^n-r^0)$ by the method of telescoping

$\displaystyle =\ a(r^n-1)$

$\displaystyle \therefore\ \sum_{k=1}^n\,T_k\ =\ \frac{a(r^n-1)}{r-1}$
Olinguito is offline  

  My Math Forum > High School Math Forum > Algebra

arithmetic, geometric, progression

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Arithmetic and Geometric progression. jiasyuen Algebra 5 April 23rd, 2014 01:42 PM
Several questions on Arithmetic and Geometric progression rnck Real Analysis 3 November 18th, 2013 09:19 PM
Arithmetic Progression jareck Algebra 3 July 6th, 2012 07:38 AM
Arithmetic progression Algebra 2 April 5th, 2012 10:46 PM
Arithmetic Progression help! Francis410 Algebra 1 March 22nd, 2011 08:02 AM

Copyright © 2019 My Math Forum. All rights reserved.