Question 1122508: how many terms of the G.P. 8,16,32,64... have their sum 8184?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The terms are all powers of 2. So one way to make the problem easy is to use the fact that the sum of the powers of 3 from 0 to (n-1) is (2^n)-1.
So let's start by adding the three terms 1, 2, and 4 to the beginning of the given sequence. The sum is now 8184+7 = 8191.
2^13 = 8192; that means our sequence with the three extra terms contains 13 terms -- from 2^0 to 2^12.
Since we added 3 terms to the given sequence, the given sequence has 10 terms.
Answer: 10 terms
And, having written that response, here is something that might make the work even easier.
We can divide every term in the given sequence by 8; then the question will be finding the number of terms in the sequence 1, 2, 4 ,8, ... that have a sum of 8184/8 = 1023.
Since 2^10 = 1024, the immediate answer is 10 terms.
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