Math sequence forumua: "If you save one penny on day one, then add..."

[fredjara]

I am trying to find a formula (if such exists) for the math addition sequence that looks something like this:
1+1=2 +2 = 4+4=8...
The problem is similar to this: "If you save one penny on day one, then add a penny to it on day two, you have saved two pennies. You then add an additional two pennies the next day (day three) and you have a total of four pennies. On day five you add four pennies to your total from day three, and you now have eight pennies...etc.
Is there a formula for this?
If so, please share. Thanks!
Fred

 

[Deleted member 4993]

I am trying to find a formula (if such exists) for the math addition sequence that looks something like this:
1+1=2 +2 = 4+4=8...
The problem is similar to this: "If you save one penny on day one, then add a penny to it on day two, you have saved two pennies. You then add an additional two pennies the next day (day three) and you have a total of four pennies. On day five you add four pennies to your total from day three, and you now have eight pennies...etc.
Is there a formula for this?
If so, please share. Thanks!
Fred

Total money on 1st. day = 1 = 2⁰

Total money on 2nd. day = 2 = 2¹

Total money on 3rd. day = 4 = 2²

Total money on 4th. day = 8 = 2³

Total money on 5th. day = 16 = 2⁴

See the pattern....

 

[fredjara]

Merged with original thread

Let's say you have a penny that you save on day one. On day two you add a penny to your previous total, leaving you with two pennies. On day three you add two pennies to your already saved two pennies leaving you with four pennies. On day four you add four pennies, leaving you with eight pennies, etc. On day 30 how many pennies would you have saved?
Is there a formula for this? I know there is a formula for 1+2+3+4...etc. Which is x(x-1)/2. But is there a formula for the problem I presented?

 

[Deleted member 4993]

Let's say you have a penny that you save on day one. On day two you add a penny to your previous total, leaving you with two pennies. On day three you add two pennies to your already saved two pennies leaving you with four pennies. On day four you add four pennies, leaving you with eight pennies, etc. On day 30 how many pennies would you have saved?
Is there a formula for this? I know there is a formula for 1+2+3+4...etc. Which is x(x-1)/2. But is there a formula for the problem I presented?

You had asked this question at:

http://www.freemathhelp.com/forum/t...-on-day-one-then-add-quot?p=384715#post384715

Did you read the answers?

 

[Ishuda]

I am trying to find a formula (if such exists) for the math addition sequence that looks something like this:
1+1=2 +2 = 4+4=8...
The problem is similar to this: "If you save one penny on day one, then add a penny to it on day two, you have saved two pennies. You then add an additional two pennies the next day (day three) and you have a total of four pennies. On day five you add four pennies to your total from day three, and you now have eight pennies...etc.
Is there a formula for this?
If so, please share. Thanks!
Fred

This is what is known as the sum of a geometric series. The first term in the series is a₀ [in your case a₀=1] and each succeeding term is r times the previous one [r, in your case, is 2]. The sum S(n) is
S(n) = a₀ [ 1 + r + r² + r³ + ... + rⁿ ] = \(\displaystyle a_0\, \frac{r^{n+1}\, -\, 1}{r\, -\, 1}\)
In your case n=30 so
S(30) = \(\displaystyle \frac{2^{30+1}\, -\, 1}{2\, -\, 1}\, =\, 2^{31}\, -\, 1\)