level flight: IAS doubled, then CL?

[bayblade]

but dynamic pressure = IAS = (1/2) * density * (TAS)^2

accordingly twice as much as increase in IAS would be caused by 1.414 times increase in TAS ( (1.414)^2 =2)

similarly, twice as much as increase in TAS would cause 4 times increase in IAS.
It satisfies the equation, so why is it wrong?

I'd understand if the original question stated 'if the TAS is doubled', then the answer would be (1/4)Cl, but I don't understand how TAS and IAS become linear variables vis a vis 'double the IAS, doubles the TAS'.