Geometric series with continuous compounding (deposits into sinking fund)

[radnorgardens]

Hi all,

Q. A regular savings of $500 is made into a sinking fund at the start of each year for 10 years. Determine the value of the fund at the end of the tenth year on the assumption that the rate of interest is 10% compounded continuously.

This is part b) of the question. I previously answered part a) which was 11% compounded annually - I answered correctly using the a((rⁿ-1)/(r-1)) formula.

So for continuous compounding, I'm using e^rt/100. With r=10, t=10. Thus, e^(10x10)/100. Plug this into the formula:

500(1.10??)((1.10e^(10x10)/100-1)/(1.10-1)) = 500(1.10??)((1.99-1)/(0.10)) = 500(1.10??)(19.9)

The 1.10, as part of 500(1.10) I know is wrong. It needs to reflect continuous compounding. In the same way that it would need to reflect monthly compounding, by dividing 10 by 12 to get 0.83, and thus 1.083. So considering it's continuous compounding, it must be smaller than this.

Now I'm stuck. The textbook answer is $9,028.14.

Again, thanks for any help.

 

[tkhunny]

If all else fails, please resort to Basic Principles.

Payment #1: \(\displaystyle 500\cdot e^{0.1\cdot 10}\)
Payment #2: \(\displaystyle 500\cdot e^{0.1\cdot 9}\)
Payment #3: \(\displaystyle 500\cdot e^{0.1\cdot 8}\)
etc...
Payment #10: \(\displaystyle 500\cdot e^{0.1\cdot 1}\)

Add them all up. \(\displaystyle 500\cdot \left(e^{1} + e^{0.9} + e^{0.8} + ... + e^{0.1}\right) = 500\cdot\dfrac{e-1}{1-e^{-0.1}}\)

It's a Geometric Series. Get GOOD at adding them up and you'll NEVER have to worry about what formula to use.

 

[radnorgardens]

If all else fails, please resort to Basic Principles.

Payment #1: \(\displaystyle 500\cdot e^{0.1\cdot 10}\)
Payment #2: \(\displaystyle 500\cdot e^{0.1\cdot 9}\)
Payment #3: \(\displaystyle 500\cdot e^{0.1\cdot 8}\)
etc...
Payment #10: \(\displaystyle 500\cdot e^{0.1\cdot 1}\)

Add them all up. \(\displaystyle 500\cdot \left(e^{1} + e^{0.9} + e^{0.8} + ... + e^{0.1}\right) = 500\cdot\dfrac{e-1}{1-e^{-0.1}}\)

It's a Geometric Series. Get GOOD at adding them up and you'll NEVER have to worry about what formula to use.

Thank you tkhunny.

I calculated the correct answer adding up the individual payments.