# Question about series and finding an expression/closed form expression

• Nov 29th 2012, 03:09 PM
bzach43
Question about series and finding an expression/closed form expression
Bill deposits 10,000 dollars every year in a savings account earning 3% a year compounded annually. He makes the first deposit at his 25th birthday. He makes deposits until his 45th birthday. After this point, he does not deposit or withdraw money from the account until he is 65 years old.

1. Let Mn be the amount of money in thousands of dollars in Bill's savings account after n years. Find an expression for M0, M2 and M3.

2. Find a closed form expression for how much money Bill has in his savings account at his 45th birthday.

3. Find a closed form expression for how much money Bill has in his savings account at is 65th birthday.

I'm having a TON of help with this, I just can't figure it out. I can figure out a recursive formula for it (Mn = 10 + .03(M(n-1)) or Mn = (1.03)(10+M(n-1)) ), but I just don't know what to do. What's the difference between an "expression" and a "closed form expression"? Is the recursive formula I figured out good enough for part 1 (if I fill it in with the appropriate years, of course)? But then what do I do for the last two parts? What's the difference there? I can't seem to figure out how to make this a geometric series, or fit any type of series formula or anything that we've been learning about.

Also, by the way, I'm new here. I'm a college freshman in calc 2, which is where this question is coming from! Thanks for any help.
• Nov 29th 2012, 03:59 PM
MarkFL
Re: Question about series and finding an expression/closed form expression
A closed-form expression will be a function that will give you a value from one calculation using an input, while a recursion requires you to begin with some initial value and then to compute all successive values up to the input.

1.) We find:

$\displaystyle M_0=10$

$\displaystyle M_1=1.03\cdot10+10=20.3$

$\displaystyle M_2=1.03(1.03\cdot10+10)+10=30.909$

2.) We may now state:

(1) $\displaystyle M_{n}=1.03M_{n-1}+10$

This is a linear inhomogeneous recursion. To find the closed-form, we may state:

(2) $\displaystyle M_{n+1}=1.03M_{n}+10$

Now we may use symbolic differencing to obtain a homogeneous recursion. Subtracting (1) from (2) we find:

$\displaystyle M_{n+1}=2.03M_{n}-1.03M_{n-1}$

Now we have a homogeneous recursion, whose associated characteristic equation is:

$\displaystyle r^2-2.03r+1.03=0$

$\displaystyle (r-1)(r-1.03)=0$

Given the characteristic roots, the closed-form is then:

$\displaystyle M_n=k_1\cdot1^n+k_2\cdot1.03^n$

$\displaystyle M_n=k_1+k_2\cdot1.03^n$

Using the initial value we computed in part 1.), we may determine that parameters $\displaystyle k_i$:

$\displaystyle M_0=k_1+k_2=10$

$\displaystyle M_0=k_1+1.03k_2=20.3$

Solving this system, we find:

$\displaystyle k_1=-\frac{1000}{3},\,k_2=\frac{1030}{3}$ hence:

$\displaystyle M_n=\frac{1}{3}\left(1030\cdot1.03^n-1000 \right)=\frac{1000}{3}\left(1.03^{n+1}-1 \right)$

Now, you may use this to find $\displaystyle M_{20}$, the amount Bill has at his 45th birthday.

3.) See if you can apply the above technique, using the result from part 2.) as your starting value, and you will begin with a homogeneous recursion.