Conjecture an expression for Sn if Sn = u1+u2+...un where Un = n*n!

M

Masaru

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I have difficulty with the following question:

The sequence of numbers {un} is defined by un=n*n!.​

1) Let Sn= u1+u2+u3+...+un. Investigate Sn for several different values of n.
2) Hence conjecture an expression for Sn.

1) is easy and can be worked out as follows:

S1 = 1*1! = 1
S2 = 1*1! +2*2! = 1 + 4 = 5
S3 = 1*1! +2*2! + 3*3! = 1 + 4 + 18 = 23
S4 = 1*1! +2*2! + 3*3! + 4*4! = 1 + 4 + 18 + 96 = 125
S5 = 1*1! +2*2! +...+5*5! = 725

However, I have no idea how I can work out 2) to make sense of these terms as a sequence because I cannot see any relation between each term in this sequence of 1, 5, 23, 125, 725... It is certainly not arithematic and it is not geometric or quadratic, either.

I would much appreciate it if someone can help me with this.

Thank you.​
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Masaru said:
The sequence of numbers {un} is defined by un=n*n!.
1) Let Sn= u1+u2+u3+...+un. Investigate Sn for several different values of n.
2) Hence conjecture an expression for Sn.

1) is easy and can be worked out as follows:
S1 = 1*1! = 1
S2 = 1*1! +2*2! = 1 + 4 = 5
S3 = 1*1! +2*2! + 3*3! = 1 + 4 + 18 = 23
S4 = 1*1! +2*2! + 3*3! + 4*4! = 1 + 4 + 18 + 96 = 125
S5 = 1*1! +2*2! +...+5*5! = 725 ​
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Click to expand...
You have two mistakes above. For corrections see HERE.

Does \(\displaystyle S_n=(n+1)!-1~?\) BUT, if so WHY?
 
Thank you for your help.

pka said:
You have two mistakes above. For corrections see HERE.

Does \(\displaystyle S_n=(n+1)!-1~?\) BUT, if so WHY?
Click to expand...

Thank you so much for your help.

Un = n*n! = n*n(n-1)(n-2)(n-3)....3*2*1 = (n+1-1)n! = {(n+1)-1}*n! = (n+1)n! - n! = (n+1)! - n!

So Sn = (2!-1!) + (3!-2!)+(4!-3!)+...+{n!-(n-1)!}+{(n+1)!-n!} = (2! + 3! +4! +...+n!) -
(2! + 3! +4! +...+n!) + (n + 1)! - 1! = (n + 1)! - 1​
 
Gets to scary large terms quickly:
1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799,
479001599, 6227020799, 87178291199, 1307674367999, 20922789887999, 355687428095999, 6402373705727999, 121645100408831999, ......

Go see it at Sloane's integer sequences; sequence #A033312
 
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