Make a conjecture about the sum? Confused on what they want exactly

[mr_coffee]

Hello everyone,

I'm confused on the directions. It says, Evalute the sum, for n = 1, 2, 3, 4, and 5. Make a conjecture about a formula for this sume for general n, and prove your conjecture by mathematical induction.

This is the sum and my work:
http://img351.imageshack.us/img351/9551/2pq2.jpg I'm having a hard time figuring out that formula, and what exactly is a conjecture? In the other readings and this one they don't say anything about a conjecture.

I was thinking that maybe they just want me to place an n where the k is?
like n/(n+1)! ?

Thanks!

 

[StatusX]

Why don't you actually add up the fractions? You should see a pattern emerge.

 

[mr_coffee]

I kind of see a pattern at the very last 2 terms,
it goes from 119/120 to 719/720, the numerator is incremented by 600 and the denominator also by 600.
But before that it goes
n = 1
1/2
n = 2
5/6
n = 3
23/24
n = 4
119/120
n = 5
719/120
n = 6

if i keep going i get
n = 7
2519/2520

Oo i see it now hah...

The denominator is +1 the numerator
but if i use n that won't work for all cases.
n/n+1, n >= 1

Is there a technique to approach conjectures? I see the pattern but i don't see how n is producing the numbers.

 

[StatusX]

The denominators are 1,2,6,24,120,... Do you recognize these?

 

[mr_coffee]

I know they are factorials, like
1! = 1
2! = 1*2 = 2
3! = 1*2*3 = 6
4! = 1*2*3*4 = 24
5! = 1*2*3*4*5 = 120

which is 1/n!

 

[mr_coffee]

I think i got it, am i allowed to write the following?

Let n be an integer, where n >= 2 such that Sumnation k = 2 to n (n!-1)/n!

THis also doesn't work i just saw, because i can't change the orginal sum, which is letting k = 1, i can't just change it can i?

Is this what a conjecture is? I'm guessing the pattern of the output and I must prove the right and left are equal?

http://img445.imageshack.us/img445/5030/lastscanrf1.jpg

 

[StatusX]

Well then just use (n+1)! instead of n!. By "conjecture" they mean find a formula that seems right (works for low numbers). Then you need to prove the formula is correct using induction.

 

[mr_coffee]

Ahh yes! it works, thanks a ton Status!