Identify the element among RLC
this looks like homework, so I'm only going to give hints


Write the expression for the total complex impedance of the circuit for all three cases (R, L  or C)
Assume a fixed voltage as input, calculate the magnitude of the current for all three cases
Identify the case, where the current actually goes up


EDIT based on the work done.

There nothing special about the number 2 here, it could 1.3, 20.5 or any other number. The only thing that's relevant is that it is larter than 1, and that means  that the current gets bigger, not smaller. 

You can simply look at the magnitude of the impedance and see if it goes up or down. Calling your inductor L0 we have

$$|Z_R| = \sqrt{R^2+(\omega L_0)^2}$$
$$|Z_L| = \omega(L+L_0)$$
$$|Z_C| = \omega L_0-1/(\omega C)$$

Compare this to the magnitude of the inductor alone, $$ \omega L_0 $$
In the first two cases, the magnitude of the impedance gets bigger, i.e. the current goes down. Only in the last case, the magnitude can get smaller (depending on the values of C and the frequency), so if the current gets bigger, it has to be a capacitor.