Growth Problem: initial value 10, 50 periods, +10 1st period, +20 2nd period, etc

[wot]

Initial value is 10
There are 50 periods.
In the first period, 10 is the added value.
Each successive period, the added value increases by 10

period 1 10 + 10 = 20
period 2 20 + 20 = 40
period 3 40 + 30 = 70
period 4 70 + 40 = 110
etc.

What is the final figure after 50 periods?
Is there an equation that can be applied?
Thanks.

 

[pka]

Initial value is 10
There are 50 periods.
In the first period, 10 is the added value.
Each successive period, the added value increases by 10
period 1 10 + 10 = 20
period 2 20 + 20 = 40
period 3 40 + 30 = 70
period 4 70 + 40 = 110 etc.

This is a recessive function: \(\displaystyle g(1)=20,~g(n+1)=10\cdot (n+1)+g(n)\)
SEE HERE

 

[Dr.Peterson]

Initial value is 10
There are 50 periods.
In the first period, 10 is the added value.
Each successive period, the added value increases by 10

period 1 10 + 10 = 20
period 2 20 + 20 = 40
period 3 40 + 30 = 70
period 4 70 + 40 = 110
etc.

What is the final figure after 50 periods?
Is there an equation that can be applied?
Thanks.

What is your context, so we can know what kinds of answers are suitable?

What have you learned about series?

Also, I'm not quite sure whether you are asking for the amount for period 50, or for the sum of all 50 amounts. The wording leaves the nature of the "values" vague. The nth individual value is 10 plus the sum of an arithmetic series, 10+(10+20+30+40+...); there is a simple formula for that.

 

[wot]

What is your context, so we can know what kinds of answers are suitable?

What have you learned about series?

Also, I'm not quite sure whether you are asking for the amount for period 50, or for the sum of all 50 amounts. The wording leaves the nature of the "values" vague. The nth individual value is 10 plus the sum of an arithmetic series, 10+(10+20+30+40+...); there is a simple formula for that.

Thank-you for both replies.
I am unsure how to apply the function equation.
I am looking the amount of period 50.
It is in relation to return on an investment. It would be 10 plus 10 plus 20 plus 30 plus 40 and so on. I could laboriously add them but I would like to see the final 50th period answer through an equation, though I am not versed in math.

 

[pka]

Initial value is 10, There are 50 periods. In the first period, 10 is the added value.
Each successive period, the added value increases by 10
period 1 10 + 10 = 20
period 2 20 + 20 = 40
period 3 40 + 30 = 70
period 4 70 + 40 = 110
etc.

My first reply was done in a bit of a hurry (I give a class on Wed) Back now.
As I said before as posted this is clearly a recursive relation. I solved it.
It turns out to be: \(\displaystyle g(n)=5n^2+5n+10\).
You can see more HERE. You can use that site for more values.

In this thread, I see a suggestion that you may want a sum. Here is that done.

 

[wot]

My first reply was done in a bit of a hurry (I give a class on Wed) Back now.
As I said before as posted this is clearly a recursive relation. I solved it.
It turns out to be: \(\displaystyle g(n)=5n^2+5n+10\).
You can see more HERE. You can use that site for more values.

In this thread, I see a suggestion that you may want a sum. Here is that done.

Thank-you for both replies.
Thank-you pka for solving it to period 50.
I posted an earlier reply yesterday but it seemingly never made it. The question was in relation to a return on investment strategy.

 

[Dr.Peterson]

Thank-you for both replies.
I am unsure how to apply the function equation.
I am looking the amount of period 50.
It is in relation to return on an investment. It would be 10 plus 10 plus 20 plus 30 plus 40 and so on. I could laboriously add them but I would like to see the final 50th period answer through an equation, though I am not versed in math.

You haven't answered my question about what you know, which was intended to inform me about how I can best help you be able to do this yourself. Subtracting the first term, this (0+10+20+...+490) is an arithmetic series, with first term 10 and common difference 10. I'll presume that you don't know anything about this; but it is easy to learn.

The formula for the sum of such a series is simply the number of terms times the average term. The average term is (0+490)/2 = 245; the number of terms is 50; so the sum is 245*50 = 12,250. Add that 10 back on, and we have 12,760.

The formula found by pka gives 5n^2 + 5n + 10 = 5*50^2 + 5*50 + 10 = 12,760. So we agree.