According to Charles' Law the volume of an ideal gas is directly proportional to its absolute temperature in Kelvin keeping the pressure constant.
V∝ T, P is constant
where V, T and P are volume, temperature and pressure
= 
where V₁, T₁, V₂ and T₂ are initial volume, initial temperature, final volume and final temperature.
For the first one the pattern is multiply the previous number by five as you see 1 x 5 = 5 and so on. To keep adding to it you would do
125 x 5 = 625 625 x 5 = 3125 3125 x 5 = 15625
Now for the second one the pattern is divide the previous number by three as you can see 2187 / 3 = 729 and so on. To keep going you would
81 / 3 = 27 27 / 3 = 9 9 / 3 = 3
I hope this helps you and if you have anymore questions i'll be glad to answer them.
It will sink below the surface. Since dish soap is a surfactant, it will alter the water's surface tension. Lowering the surface tension, the water will no longer be able to support the mass of the twig.
On a molecular level there is a lot of movement which in turn is the reason why heat is generated.
