I agree with the 7.97. I checked my Excel table and saw an error. I attached a correct table here. It shows during the 8th day. Sorry.

In the problem, the number of drops for the first three days are 2, 4, and 8.

Notice that the number of drops forms a sequence. So, our next step is to determine if it is an arithmetic or geometric sequence. Since 2, 4 and 8 are all divisible by 2, then it is possible that this is a geometric sequence. To verify, determine if the consecutive numbers have common ratio.

To do so, apply the formula:

`r =a_n/a_(n-1)`

`r=a_3/a_2 =8/4=2`

`r=a_2/a_1=4/2=2`

Since the value of r's are the same, hence, the number of drops form a geometric sequence.

That means, to determine nth day when 500th rain drops on the leak, apply the formula of sum of geometric sequence.

`S_n=(a_1(1 - r^n))/(1-r)`

So, plug-in `S_n=500` , `a_1=2` and `r=2` .

`500=(2(1-2^n))/(1-2)`

Then, simplify the equation.

`500=(2(1-2^n))/(-1)`

`500=-2(1-2^n)`

`-250=1-2^n`

`-251=-2^n`

`251=2^n`

And, take the natural logarithm of both sides to remove the n in the exponent.

`ln 251=ln2^n`

`ln251=nln2`

`(ln251)/(ln2)=n`

`7.97=n`

Rounding off to the nearest whole number, the value of n becomes:

`8=n`

**Hence, the 500th rain drops on the 8th day.**

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