The sum of a geometric series is where a(1) is the first term, r is the common factor, and n is the nth term (the number of the term that = the sum). Set this = 500 and solve for n. The n needed (the numbered term) will be greater than that n.

a(1) = 2 and r=2. The desired sum is 500.

Recall that when log (base x)y = n so log(base 2) 251 = n = approx. 7.97

Therefore the sum of the first 7.97 terms is 500, so the sum exceeds 500 with the 8th term or 8th day.

You could have also solved this by writing out the amount he saved each day, and adding: 2, 4, 8, 16, 32, 64, 128, 256 (the sum is $510, which exceeds $500 on the 8th day.)

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