Sum of i

Date: 03/23/2002 at 09:57:56
From: Julie Gal
Subject: Summation

If the sum as i goes from 1 to n of 2^i is 2^n -1, what is the sum
as i goes from 1 to n of 3^i ?  

Thanks.  
Julie Gal

Date: 03/23/2002 at 12:22:52
From: Doctor Jubal
Subject: Re: Summation

Hi Julie,

Thanks for writing Dr. Math.

I started by writing down the first few sums, and here's what I got

  n     3^n     sum (3^i)
  0     1       1
  1     3       4
  2     9      13
  3    27      40
  4    81     121
  5   243     364
  6   729    1093

It appears to me, that in each case, the sum is one less than half of 
3 to the next highest power:

  1 = (3-1)/2
  4 = (9-1)/2
 13 = (27-1)/2
 40 = (81-1)/2 etc.

So, let's see if we can prove this: sum from 1 to n of 3^i = 
(3^(n+1)-1)/2.  We'll do it by mathematical induction.

First, the basis step.  For n=0, 1 = (3^1 - 1)/2 = 1.

Now, the induction step. If we suppose that the statement is true for 
some value of n, does it necessarily follow that it is also true for 
the next highest value of n?

Since the nth sum is (3^(n+1) - 1)/2, if we add the next term to it, 
we get

  (3^(n+1) -1)/2 + 3^(n+1) = 
  (3/2)*3^(n+1) - 1/2 = 
  (3* 3^(n+1) - 1) / 2 = 
  (3^(n+2) - 1)/2

Which is exactly what our proposed formula would predict for the value 
of the sum from i=1 to n+1.

So, if the formula is true for some value of n, it must also be true 
for all greater values of n. However, we've already observed that it 
is true for n=0, so we can conclude that it is also true for all 
postive n.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/23/2002 at 22:10:44
From: Julie Gal
Subject: Summation

Dear Dr Jubal:  

Thank you for your solution to my question. I followed your proof.  
Your summation is from  i = 0 to n instead of  i = 1 to n, which of 
course also includes the number 1. My math 12 honors class is studying 
sequences and series and that would be a great extra credit question 
to have them work on. 

Sincerely, 
Julie Gal