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Jed does pushups every week day. On Monday he does $7.$ He doubles his average every day he works out. How many pushups does he do next Monday?

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  • $\begingroup$ What do you mean by "doubles his average" ? Do you mean that on Tuesday he does $7/1*2=14$, on Wednesday he does $((7+14)/2)*2$ ? $\endgroup$ – Stefan Hamcke Jan 30 '14 at 15:40
  • $\begingroup$ Doubles the average number of push ups he has done. Just a question I tested on a year 7 class. Got kind of addicted to this and ran out of questions I needed the answer to $\endgroup$ – Plastic Fork Jan 30 '14 at 15:44
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I'm going to try to first define the question more closely and then frame it more mathematically, for the lolz:

Question

Every weekday, Jed does some pushups. On the first day we'll consider, which is a Monday, Jed does $7$ pushups. Every weekday after that, Jed does some number of pushups such that the average number of pushups he's done per day since (and including) that first Monday is double this same quantity as computed the previous day.

So, for example, on Monday, this average is $7$. On Tuesday, Jed needs to do $21$ pushups so that this average is $\frac{7 + 21}{2} = 14$.

How many pushups must Jed do the Monday after the first Monday to satisfy this rule?

Rephrasing

Consider the sequence $s(n)$, representing the number of pushups Jed does on the weekday which is $n-1$ weekdays after the first Monday.

If we define the sequence $S$ as:

$$ S(n) := \frac{\sum_{k=1}^{n} s(k)}{n} $$

Then we require that $s$ is a sequence such that ($\dagger$):

$$ s(1) = 7 $$ $$ \forall n > 1, S(n) = 2S(n-1) $$

  1. Can you provide (and prove) a general formula for the value of $s(n)$?
  2. What is the value of $s(6)$ (the next Monday)?
  3. Finally, can you prove that there is only one sequence $s$ satisfying $\dagger$?
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    $\begingroup$ Thats what I said $\endgroup$ – Plastic Fork Jan 30 '14 at 16:16
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On Monday (day $1$) he does $7=7*1$. Let's assume by induction that till day $n$ he did $7*m$ on day $m$ for each $m\le n$. Then the average is $$\left(\sum_{m=1}^n 7*m\right)/n=7/n*\sum_{m=1}^n m=\frac{7n(n+1)}{2n}$$ Since he doubles that average, he does $7(n+1)$ on day $n+1$. This completes the induction.

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EDIT:

Taking "doubling his average" into account, rather than "average previous days, then double," we have $7$ on Monday, then an average of $14$ between Monday and Tuesday which makes $21$ for Tuesday's count, then $28$ for the average of Monday-Wednesday making Wednesday's count be $2*28=56$, then $56$ for the average of Monday-Thursday making Thursday's count be $5*28=140$ then $112$ for the average of Monday-Friday making Friday's count be $560-224=336$.

On Monday he doubles his average again. What is his average? If Saturday and Sunday are discounted, we double $112$ to get $224$ making Monday's count be $6*224-560=784$.

If Saturday and Sunday are not workout days, but are averaging days, then the previous average $112$ drops to $80$ and the new average for Monday is $160$, making Monday's count be $8*160-560=720$.

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From wikipedia: "A weekday is any day of the week except Sunday and often also Saturday." So he doesn't work out on Sunday, and it's not clear if he works out on Saturday. It is also not clear whether the days where he doesn't work out are counted in the average.

So there are n days where he works out, and n is either 7 or 6. We calculate the average over m days, where m is either 8 or n.

The average on the first day is 7, and on the n-th day it is 7 * 2^(n - 1). The total on Monday is m * 7 * 2^(n - 1).

The average on the last workout day before Monday is 7 * 2^(n - 2). The total on the last workout day before Monday is (n - 1) * 7 * 2^(n - 2).

The pushups on the last monday are the difference between both totals, which is

  7 * 2^(n - 2) * (2m - n + 1). 
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  • $\begingroup$ Wait, how many pushups is he supposed to do on Wednesday? $\endgroup$ – abiessu Jan 30 '14 at 18:07

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