$\textbf{Calculate value of $\mathbf{x}$ in }$\\\\ \mathbf{\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\left[\frac{x}{3!}\right]+.................+\left[\frac{x}{2007!}\right] = 1005}$\\\\ \textbf{Where $\mathbf{\left[ . \right]=}$ Greatest Integer function.}$
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3 Answers
1[x/1!]+[x/2!]+...[x/2007!]=1005
=[([x]+{x})/1!] +[([x]+{x})/2!]+...[([x]+{x})/2007!]
=[[x]/1!]+[[x]/2!]...[[x]/2007!]=1005 (1)
if [x]=k*n! ,k<n+1,
we see [x] should clearly lie between 5! and 6!
so k<6
n=5.
we have [x]=k*5!
substitute it in (1)
find the nearest rounded integer k to get [x].
cheers!
