##### what square will have more than \$1,000,000?

 Mathematics Tutor: None Selected Time limit: 1 Day

place one dollar on first square, 2 dollars on the second square, 4 on the third square, 8 on the forth square etc. on what square will more than \$1,000,000 be placed?

Sep 9th, 2015

This is an example of a geometric progression. In each square you will be doubling the value of the previous square. We can do this either by hand:

Square 1 = \$1

Square 2 = \$2*1 = \$2

Square 3 = \$2*2 = \$4

Square 4 = \$4*2 = \$8

Square 5 = \$8*2 = \$16

Square 6 = \$16*2 = \$32

Square 7 = \$32*2 = \$64

Square 8 = \$64*2 = \$128

Square 9 = \$128*2 = \$256

Square 10 = \$256*2 = \$512

Square 11 = \$512*2 = \$1024

Square 12 = \$1024*2 = \$2048

.....

But as you can see, this can take a quite long time. Another way we can look at this, is to write an equation that describes this. So at the N-th Square, we will have 2^N (that is 2 raised to the N-th power), and we need to find the value of N such that 2^N > 1,000,000

Now there is an operation that is the inverse of exponentiation, and that is the logarithm.

If we take the logarithm (with base 2) of both sides of the equation we get:

N > log-base-2(1,000,000)

N > 19.93

Since we only have integer number of squares, we know that we will have more than 1,000,000 in the 20th square.

Just to check 2^20 = 1,048,576

Please let me know if you need any clarification. Always glad to help!
Sep 9th, 2015

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Sep 9th, 2015
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Sep 9th, 2015
May 24th, 2017
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