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

Hi there! Thank you for the opportunity to help you with your question!

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

Did you know? You can earn $20 for every friend you invite to Studypool!
Click here to
Refer a Friend
...
Sep 9th, 2015
...
Sep 9th, 2015
May 24th, 2017
check_circle
Mark as Final Answer
check_circle
Unmark as Final Answer
check_circle
Final Answer

Secure Information

Content will be erased after question is completed.

check_circle
Final Answer