GHogg
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A king in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chess board. On the second square the King would place two grains of wheat, on the third square, four grains of wheat, and on the fourth square eight grains of whea. If the amount of wheat is doubled in this way on each of the remaining squares, how many grains of wheat should be placed on square 13? Also find the total number of grains of wheat on the board at their total weight in pounds. (Assume that each grain of wheat weighs 1/7000 pound)

Respuesta :

part A)

[tex]\bf n^{th}\textit{ term of a geometric sequence}\\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ n=13\\ a_1=1\\ r=2 \end{cases} \\\\\\ a_{13}=1\cdot 2^{13-1}\implies a_{13}=2^{12}\implies a_{13}=4096[/tex]

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[tex]\bf \qquad \textit{Amount for Exponential change}\\\\ A=P(1\pm r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{starting amount}\to &1\\ r=rate\to 100\%\to \frac{100}{100}\to &1.00\\ t=\textit{elapsed period}\to &13\\ \end{cases} \\\\\\ A=1(1+1)^{13}\implies A=2^{13}\implies A=8192 \\\\\\ 8192\cdot \cfrac{1}{7000}lb\implies \cfrac{8192}{7000}lb\approx 1.1702857lbs[/tex]
jorel1380
2^0=.
2^1=2
2^2=4
.
.
.
2^12=4096 grains of wheat on square 13
After 64 days:
Total wheat=2^63 x 2=2^64=18446744073709551616 grains
Weight=18446744073709551616/7000=2,635,249,153,387,078.8 lbs
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