Factorials

Algebra Level 2

1 + 2 × 2 ! + 3 × 3 ! + ⋯ + 9 × 9 ! = ? 1 + 2\times2! + 3\times3! + \cdots + 9\times9! = \, ?


The answer is 3628799.

Samer Ayoub by samer Ayoub , 46, Canada

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3 solutions

Mateus Gomes
Feb 12, 2016

∑ n = 1 9 n × n ! = 1 + 2 × 2 ! + 3 × 3 ! + 4 × 4 ! + 5 × 5 ! + 6 × 6 ! + 7 × 7 ! + 8 × 8 ! + 9 × 9 ! \Large\sum_{n=1}^{9}n\times n!=1+2\times2!+3\times3!+4\times4!+5\times5!+6\times6!+7\times7!+8\times8!+9\times9! ∑ n = 1 9 n × n ! = 1 + ( 3 − 1 ) × 2 ! + ( 4 − 1 ) × 3 ! + ( 5 − 1 ) × 4 ! + ( 6 − 1 ) × 5 ! + ( 7 − 1 ) × 6 ! + ( 8 − 1 ) × 7 ! + ( 9 − 1 ) × 8 ! + ( 10 − 1 ) × 9 ! \Large\sum_{n=1}^{9}n\times n!=1+(3-1)\times2!+(4-1)\times3!+(5-1)\times4!+(6-1)\times5!+(7-1)\times6!+(8-1)\times7!+(9-1)\times8!+(10-1)\times9! ∑ n = 1 9 n × n ! = 1 + 3 ! − 2 ! + 4 ! − 3 ! + 5 ! − 4 ! + 6 ! − 5 ! + 7 ! − 6 ! + 8 ! − 7 ! + 9 ! − 8 ! + 10 ! − 9 ! = 10 ! − 1 \Large\sum_{n=1}^{9}n\times n!=1+3!-2!+4!-3!+5!-4!+6!-5!+7!-6!+8!-7!+9!-8!+10!-9!=\large\color{#3D99F6}{\boxed{10!-1}}

16 Helpful 1 Interesting 3 Brilliant 0 Confused
Sayan Das
Aug 24, 2016

Sorry,i don't know LaTeX.Let,u(r) be a term of the series.then u(r)=r×r!=(r+1-1)×r!=(r+1)!-r!=v(r+1)-v(r). where v(r)=r! Then the sum is S(9)=v(9+1)-v(1)[By method of difference] S=10!-1=3628799.

1 Helpful 1 Interesting 0 Brilliant 0 Confused

I will teach you how to use LaTeX .

First, use \ ( , and \ ) .( You must write \, and ( together. Of course, \, and ) too.)

Next, if you want to use a fraction, then use this command, \frac{numerator}{denominator}.

Then it will appear like 1 1 \frac { 1 } { 1 } .

Second, if you want to use the greek letters, then use \alpha, \Alpha, \beta, \Beta, etc.

Then α , A , β , B , ⋯ \alpha, \Alpha, \beta, \Beta, \cdots .

If you want to know more, use this site.

LaTeX Wikipedia Mathematics .

And if you want to know how to use links, then search about this site.

Brilliant Formatting .

. . - 3 months, 3 weeks ago

Lol, the solution that he wrote was written 5 years ago. You teaching him/her latex now. Great

SRIJAN Singh - 2 weeks, 2 days ago

It's not now.

It's exactly 3 months ago.

. . - 2 weeks, 2 days ago
Samer Ayoub
Feb 12, 2016

1+2×2!+3×3!+4×4!+5×5!+ ………………+ 9×9!= -1+2+2×2!+3×3!+4×4!+5×5!+ ………………+ 9×9!= -1+3×2!+3×3!+4×4!+5×5!+ ………………+ 9×9!= -1+3!+3×3!+4×4!+5×5!+ ………………+ 9×9!= -1+4×3!+4×4!+5×5!+ ………………+ 9×9!= -1+4!+4×4!+5×5!+ ………………+ 9×9!= -1+5×4!+5×5!+ ………………+ 9×9!= -1+5!+5×5!+ ………………+ 9×9!= -1+9!+9×9!= -1+10×9!= 10!-1=3628799

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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