Question

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

Answer 1

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\\\\`