## Selena2345 3 years ago A Goblin Miner earns 1 cent the first day, 2 cents the second day 4 cents the third day, 8 cents the fourth day and so on doubling the amount each day. How much will the miner earn in 7 days ? 2 Weeks? 1 Month ( Make An Estimate Before Solving) Show/ Explain All Work

1. ChristineToTheRescue

So...

2. ChristineToTheRescue

Um, I'm not so sure how to put this in an equation...

3. Selena2345

it is not an equation

4. ChristineToTheRescue

But, you can estimate obviously that it's about 70

5. Selena2345

okay

6. ChristineToTheRescue

You need an esquation to solve this...i think

7. ChristineToTheRescue

Imma work this out first. (:

8. ChristineToTheRescue

We see that the series whose sum gives the total number of cents follows a regular pattern: each new term added to it is a power of two. This is an example of a geometric series.

9. ChristineToTheRescue

In our case, we are told that the number of cents given each day is double the number given the day before, which suggests that the ratio of this series is 2.

10. ChristineToTheRescue

no. pennies given on second day 2 ------------------------------- = 2 no. pennies given on first day 1 no. pennies given on third day 4 ------------------------------- = 2 no. pennies given on second day 2 no. pennies given on fourth day 8 ------------------------------- = 2 no. pennies given on third day 4

11. ChristineToTheRescue

obviously, the ratio of the geometric series that gives the total number of cents on a particular day is 2.

12. ChristineToTheRescue

we can use the fact that the sum of a geometric series (called S) with n terms whose ratio is r is this S = (first term)(1-r^n)/(1-r)

13. ChristineToTheRescue

I think... ><'' not very expertise on this but had hw on this like 2 wks ago.

14. ChristineToTheRescue

This means that for our series with a first term of 1 and a ratio of 2, we find the sum after n days = 1(1 - 2^n) / (1 - 2) = - (1 - 2^n) = 2^n - 1

15. Selena2345

are we done with the question christine

16. xcrypt

The equation would be: 2^x-1, where x represents the day. substitute 7 for x and you have your answer

17. ChristineToTheRescue

1 2^1-1 = 2-1 = 1 2 2^2-1 = 4-1 = 3 3 2^3-1 = 8-1 = 7 4 2^4-1 = 16-1 = 15

18. xcrypt

I am sorry, 2^(x-1)

19. ChristineToTheRescue

there you go x3

20. ChristineToTheRescue

So...basically do the same for each day doubling it

21. Selena2345

is the question whole finish

22. ChristineToTheRescue

5 2^5-1 = 18-1 = 17 and so forth

23. Selena2345

is the answer finish now christine

24. Selena2345

thank you christine