Question

In a classic math problem, a king wants to reward a knight who has rescued him from an attack. The king gives the knight a chessboard and plans to place money on each square. He gives the knight two options. Option 1 is to place a thousand dollars on the first square, two thousand on the second square, three thousand on the third square, and so on. Option 2 is to place one penny on the first square, two pennies on the second, four on the third, and so on. Think about which offer sounds better and then answer the question. Find the amount of money placed on the 20th square of the chessboard and the total amount placed on the squares 1 through 20 for each option. Option 1. Option 2.

Solution

Verified

Step 1

1 of 2

Use the formulas from Item 2: \

(a)\textbf{(a)}

Use n=20n=20 to find the amount of money placed on the 20th square of the chessboard:

an=1000na_n=1000n

a20=1000(20)a_{20}=1000(20)

a20=$20000\color{#c34632}{a_{20}=\$20000}

The total amount placed on squares 1 through 20 is:

Sn=500n2+500nS_n=500n^2+500n

S20=500(20)2+500(20)S_{20}=500(20)^2+500(20)

S20=200000+10000S_{20}=200000+10000

S20=$210000\color{#c34632}{S_{20}=\$210000}

(b)\textbf{(b)}

Use n=20n=20 to find the amount of money placed on the 20th square of the chessboard:

an=0.01(2)n1a_n=0.01(2)^{n-1}

a20=0.01(2)201a_{20}=0.01(2)^{20-1}

a20=0.01(2)19a_{20}=0.01(2)^{19}

a20=0.01(524288)a_{20}=0.01(524288)

a20=$5242.88\color{#c34632}{a_{20}=\$5242.88}

The total amount placed on squares 1 through 20 is:

Sn=0.01(2n1)S_n=0.01(2^n-1)

S20=0.01(2201)S_{20}=0.01(2^{20}-1)

S20=0.01(1048576)S_{20}=0.01(1048576)

S20=$10485.76\color{#c34632}{S_{20}=\$10485.76}

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