In a well known story the inventor of the game of chess was asked by his well king what reward he desired. "oh, not much, your majesty", the inventor responded, "just place a grain of rice on the first square of the board, 2 on the next, 4 on the next, and so on, twice as many on each square as on the preceding one. i will give this rice to the poor." (for the uninitiated, a chess board has 64 squares.) the king thought this a modest request indeed and ordered the rice to be delivered.

let f(n) denote the number of rice grains placed on the first n squares of the board. so clearly, f(1)=1, f(2)=1+2=3, f(3)=1+2+4=7, and so on. how does it go on? compute the next two values of f(n): f(4)= 15 f(5)= 31 ponder the structure of this summation and then enter an algebraic expression that defines f(n)= as a function of n.

supposing that there are 25,000 grains of rice in a pound, 2000 pounds in a ton, and 6 billion people on earth, the inventor's reward would work out to approximately tons of rice for every person on the planet. clearly, all the rice in the kingdom would not be enough to begin to fill that request. the story has a sad ending: feeling duped, the king caused the inventor of chess to be beheaded.

f(6) = 1+2+4+8+16+32 = 63

f(7) = 63 + 64 = 127

This is a geometric sequence  where r the common ratio = 2.

f(n) = a1 * (r^n - 1) / (r - 1)

here a1 = 1 and r = 2 so

f(n) = (2^n - 1) / (2 -1)

= 2^n - 1.

When n = 64 ( the last square on the chess board)

f(n) = 2^64 - 1 = 2^64 ( I think we can neglect the 1!!)

That is the total number of grains of rice.

In tons of rice this is 2^64 /  (25,000 * 2000)

= 3.689349 * 10^11 tons  approximately.

So for each person the amount of rice would be 3.689349 * 10^11 / 6 * 10^9

= 61.5 tons approx. (Answer).

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: