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### Sum of i

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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.

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
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Associated Topics:
High School Imaginary/Complex Numbers
High School Sequences, Series

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