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6 × 6! + 7 × 7! + 8 × 8! + 9 × 9! + 10 × 10! + 11 × 11! + 12 × 12! + 13 × 13! + 14 × 14! = a! - b!

 

 

a + b= ?

 Oct 7, 2017
 #1
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0

6 × 6! + 7 × 7! + 8 × 8! + 9 × 9! + 10 × 10! + 11 × 11! + 12 × 12! + 13 × 13! + 14 × 14! = a! - b!

a! - b! =[(14+1)! - 6!]

a + b =15 + 6 =21

 Oct 7, 2017
 #2
avatar+99350 
+1

How did you do that guest?

Melody  Oct 8, 2017
 #3
avatar+27558 
+3

Here are some of the intermediate steps Melody:

 

.

Alan  Oct 8, 2017
 #4
avatar+99350 
+3

Thanks Alan :)

Melody  Oct 8, 2017
 #5
avatar+21860 
+3

6 × 6! + 7 × 7! + 8 × 8! + 9 × 9! + 10 × 10! + 11 × 11! + 12 × 12! + 13 × 13! + 14 × 14! = a! - b!

a + b= ?

 

\(\begin{array}{|rcll|} \hline && n\times n! \\ &=& (n+1-1)\times n! \\ &=& [(n+1)-1]\times n! \\ &=& (n+1)\times n! -1\times n! \\ &=& n!\times (n+1) -n! \\ &=& (n+1)! -n! \\\\ &&\mathbf{ n\times n! = (n+1)! -n! } \\ \hline \end{array}\)

 

\(\begin{array}{|rcrcr|} \hline 6\times 6! &=& 7!- 6! \\ 7\times 7! &=& 8!- 7! \\ 8\times 8! &=& 9!- 8! \\ 9\times 9! &=& 10!- 9! \\ 10\times 10! &=& 11!-10! \\ 11\times 11! &=& 12!-11! \\ 12\times 12! &=& 13!-12! \\ 13\times 13! &=& 14!-13! \\ 14\times 14! &=& 15!-14! \\ \hline \text{sum} &=& 7! &\mathbf{-}& \mathbf{6!} \\ &+& 8! &-& 7! \\ &+& 9! &-& 8! \\ &+& 10! &-& 9! \\ &+& 11! &-& 10! \\ &+& 12! &-& 11! \\ &+& 13! &-& 12! \\ &+& 14! &-& 13! \\ &+& 15! &-& 14! \\\\ &=& \not{ {\color{green}7!}} && -6! \\ && {\color{red}{-}}\not{ {\color{red}7!}} && {\color{green}{+}} \not{ {\color{green}8!}} \\ && {\color{green}{+}}\not{ {\color{green}9!}} && {\color{red}{-}}\not{ {\color{red}8!}} \\ && {\color{red}{-}}\not{ {\color{red}9!}} && {\color{green}{+}} \not{ {\color{green}10!}} \\ && {\color{green}{+}} \not{ {\color{green}11!}} && {\color{red}{-}}\not{ {\color{red}10!}} \\ && {\color{red}{-}}\not{ {\color{red}11!}} && {\color{green}{+}} \not{ {\color{green}12!}} \\ && {\color{green}{+}} \not{ {\color{green}13!}} && {\color{red}{-}}\not{ {\color{red}12!}} \\ && {\color{red}{-}}\not{ {\color{red}13!}} && {\color{green}{+}} \not{ {\color{green}14!}} \\ && \mathbf{+15!} && {\color{red}{-}}\not{ {\color{red}14!}} \\\\ &\mathbf{=}& \mathbf{15! - 6!} \\ \hline \end{array} \)

 

so a = 15 and b = 6

a+b = 15+6 = 21

 

laugh

 Oct 10, 2017

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