Nth Partial Sum

**BioGrl** · November 25th 2009, 07:46 PM

I start a savings account with $200 and the account earns 6% interest per year compounded annually. Every year I add an additional $200 to the account. What is the infinite series that calculates the total value of the account after n years? (at n=0 the value is $200) What is the equation for the nth partial sum (Sn)?

**redsoxfan325** · November 25th 2009, 10:21 PM

Originally Posted by BioGrl

I start a savings account with $200 and the account earns 6% interest per year compounded annually. Every year I add an additional $200 to the account. What is the infinite series that calculates the total value of the account after n years? (at n=0 the value is $200) What is the equation for the nth partial sum (Sn)?

The initial $200$ is multiplied by $1.06^n$ . The second $200$ is multiplied by $1.06^{n-1}$ and so on

$s_0=200$
$s_1=200+1.06\cdot200$
$s_2=200+1.06\cdot200+1.06^2\cdot200$
...
$s_n=200\sum_{k=0}^n 1.06^k$

It's a geometric series.

**BioGrl** · November 26th 2009, 08:14 AM

Thanx! But I have a question. Why wouldn't Sn be raised to k-1 instead of k? I am confused on how you got to the final geometric series.
Also, if I wanted to find the value of the account after 10 years how would I do that?
Thanx sooo much!!

**redsoxfan325** · November 26th 2009, 05:47 PM

Originally Posted by BioGrl

Thanx! But I have a question. Why wouldn't Sn be raised to k-1 instead of k? I am confused on how you got to the final geometric series.
Also, if I wanted to find the value of the account after 10 years how would I do that?
Thanx sooo much!!

It should be $k$ . If you look at the pattern going up, you can see that the highest exponent in the sum $s_k$ of $1.06$ is $k$ .

You can prove this by induction if you want.

If you want to find the value after 10 years, put in $n=10$ .

**BioGrl** · November 28th 2009, 06:24 PM

Thanx!!!!

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