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If 1/8! + 1/9! = X/10! find the value of X.​

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3 votes

Answer:

x = 100

Explanation:

We are given


(1)/(8!) + (1)/(9!) = (x)/(10!)\\\\\\\textrm{For any number n, $n! = n \cdot (n-1) ..........2\cdot 1 $}\\\\n! = n \cdot (n-1)!


\textrm{By the identity that $n! = n \cdot (n-1)!$}


9! = 9 \cdot 8!\\\\\textrm{and }\\\\10! = 10 \cdot 9!\\\\


(1)/(8!) + (1)/(9!)\\\\= (1)/(8!) + (1)/(9 \cdot 8!) \\\\

The LCM of 8! and 9 · 8! is 9 · 8! since 9 · 8! can be divided by both 8! and 9 · 8!

Multiplying numerators by 9 · 8! and using 9 · 8! as the common denominator gives us


(1\cdot 9 \cdot 8!)/(8!) + (1 \cdot 9 \cdot 8!\\ )/(9 \cdot 8!) \\

=
(9 + 1)/(9 \cdot 8!) = (10)/(9\cdot 8!)\\\\\\= (10)/(9!) \;\;\;\;\;\;\textrm{ since $9\cdot 8! = 9!$}

This is given to be
(x)/(10!)

So


(10)/(9!) = (x)/(10!)\\

Switching sides,

\rightarrow (x)/(10!) = (10)/(9!)\\\\\rightarrow x = (10\cdot 10!)/(9!) }\\\\\textrm{But $10! = 10 \cdot 9!$ since $n! = n \cdot (n-1)!$}\\\\

So


x = (10 \cdot 10 \cdot 9!)/(9!) = 10 \cdot 10 = 100\\\\

User Ulli Schmid
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