# Math Help - Nth Partial Sum

1. ## Nth Partial Sum

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)? 2. 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.

3. 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!!

4. 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 $n=10$.

5. Thanx!!!!