SOLUTION: A contestant on a game show wins one chip for answering the first question correctly, two chips for answering the second question correctly, four chips for the third, eight chips f

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Question 588569: A contestant on a game show wins one chip for answering the first question correctly, two chips for answering the second question correctly, four chips for the third, eight chips for the fourth, etc. The contestant can stop anytime, but if he or she misses a question, all the accumulated chips are lost. The chips can be cashed in at the end of the game for prizes. If a contestant wishes to win a prize worth 225 chips, how many questions must the contestant answer correctly?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The number of chips is in geometric sequence (or geometric progression) where each term is 2 times the previous one (common ratio=2).
The nth question is worth 2%5En chips.
If the contestant answers the first n questions correctly, the number of accumulated chips will be the sum, S:
S=1+2+2%5E2+2%5E3+ ..... +2%5E%28n-1%29+2%5En
Textbooks have formulas to calculate such sums, but I'd rather open my mind than a textbook.
Multiplying times 2:
2S=2+2%5E2+2%5E3+ ..... +2%5E%28n-1%29+2%5En+2%5E%28n%2B1%29
and subtracting the expression for S from the expression for 2S:
S=2S-S=2%5E%28n%2B1%29-1
2%5E7=128 and 2%5E8=256
so the contestant has to aim for n+1=8, and answer the first 7 questions correctly.