document.write( "Question 515493: the total number of dollars you would have received if you had received 1 dollar on your first birthday, 2 on your second birthday 4 on your third birthday and continued doubling the dollars on each birthday to your 18 \n" ); document.write( "

Algebra.Com's Answer #344035 by drcole(72) $\"\"$ $\"About$

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Notice the pattern. On your first birthday, you received $\"1+=+2%5E0\"$ dollars. On your second birthday, you received $\"2+=+2%5E1\"$ dollars. On the third birthday, you received $\"4+=+2%5E2\"$ dollars. The pattern continues, with the amount doubling each year. So on your kth birthday, you receive $\"2%5E%28k+-+1%29\"$ dollars (notice that the exponent is always one less than the number of the birthday). This means that the total amount you receive from your first birthday to your eighteenth is:\r
\n" ); document.write( "\n" ); document.write( "2^0 + 2^1 + 2^2 + ... + 2^16 + 2^17 dollars (since the amount you will receive on your eighteenth birthday will be $\"2%5E%2818+-+1%29+=+2%5E%2817%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "You could add up all eighteen numbers, but there is an easier way. In general, if we have a sum of the form:\r
\n" ); document.write( "\n" ); document.write( "r^0 + r^1 + r^2 + ... + r^(N - 2) + r^(N - 1)\r
\n" ); document.write( "\n" ); document.write( "then the sum is given by the formula:\r
\n" ); document.write( "\n" ); document.write( " $\"+%28r%5EN+-+1%29%2F%28r+-+1%29\"$ \r
\n" ); document.write( "\n" ); document.write( "In the case of this problem, r = 2 and N = 18, so we have a sum of:\r
\n" ); document.write( "\n" ); document.write( " $\"+%282%5E%2818%29+-+1%29%2F%282+-+1%29+=+%28262144+-+1%29%2F%282+-+1%29+=+262143%2F1+=+262143\"$ dollars. \n" ); document.write( "

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