1. ## Need Help Quick

it is not the hardest puzzle but I need to know a way to figure it out without doing 50 sums, the most important thing is the way it's figured out and no really hard math answers!

You are washing up for your mum everyday, on the first day you make 1p, on the next day 2p on the next day 4p, so the amount you get is doubled everyday, so the day after it would be 8p then 16p etc

how much do you earn in 1 week, 2 weeks, 3 weeks and one month, remember I need to know how it is done and simplely like using powers of and stuff.

2. First day you get 1 cent (2^0)
2nd day you get 2 cents (2^1)
3rd day you get 4 cents (2^2)
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.
.
8th day (which is one week later from the first day) you get (2^7) cents
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.
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etc etc etc

You are washing up for your mum everyday.
On the first day you make 1p, on the next day 2p, on the next day 4p,
so the amount you get is doubled everyday.
So the day after it would be 8p, then 16p, etc.

How much do you earn in 1 week? .2 weeks? .3 weeks? .one month?

On successive days you earn: .1, 2, 4, 8, 16, 32, ...

By the $n^{th}$ day, you've earned a total of:
. . $S_n \;=\;1 + 2 + 2^2 + 2^3 + \cdots + 2^{n-1}$

This is a geometric series with: first term $a = 1$, common ratio $r= 2$, and $n$ terms.

Its sum is: . $S_n \;=\;1\cdot\frac{2^n-1}{2 -1} \;=\;2^n - 1$

One week: . $S_7 \;=\;2^7 - 1 \;=\;127\,p$

Two weeks: . $S_{14} \;=\;2^{14}-1 \;=\;16,383\,p$ . . . . etc.