$\dfrac{1}{10!} = a, \space \space 1+ \dfrac{1}{10! + 11!}$

Question

$$\dfrac{1}{10!} = a, \space \space 1+ \dfrac{1}{10! + 11!}$$

It wants me to evaluate this expression in terms of $a$. Let me show my work as illustrated below

$$ 1+ \dfrac{1}{10! + 11!}$$

After factoring we get

$$\dfrac{1}{10!} \biggr( 1+ \dfrac{1}{11}\biggr ) + 1$$

I couldn't proceed further. Any help will be appreciated.

Answer 1

No, it should be $1+a/12$ because $(10!+11!)/10!=1+11=12$.

Answer 2

from $\dfrac{1}{10!} = a$, can we can conclude that $\dfrac{1}{a} = 10!$.

Use this to see that:

$$1 + \dfrac{1}{10! + 11!} = 1 + \dfrac{1}{\dfrac{1}{a} + 11!}$$

$$=1 +\dfrac{1}{\dfrac{1}{a} + \dfrac{11!a}{a}}$$ $$=1 +\dfrac{1}{\dfrac{1+11!a}{a}}$$ $$=1 +\dfrac{a}{1+11!a}$$ $$=1 +\dfrac{a}{1+11!\cdot\dfrac{1}{10!}}$$ $$=1 +\dfrac{a}{12}$$

Can you go on from here?

Answer 3

Given $$\dfrac{1}{10!} = a$$

$$1+ \dfrac{1}{10! + 11!}$$ $$1+ \dfrac{1}{10! + 11\cdot 10!}$$ $$1+ \dfrac{1}{10!(12)}$$ $$1+ \dfrac{a}{12}$$