Hi,

I've been impressed with the Manhattan challenge problem series, so here's my two cents. I came across this problem today while calculating the return for my investment :)

Question: I plan to buy an investment instrument for a 20 years term. The investment company guarantees 10% annual rate of return compounded yearly. If I invest $20,000 per year throughout the term period (i.e. 20 years), what's the total fund value at the end of term period?

Solution:

Compound Interest formula = P*((1 + r/100)^n), where P is principal, r is rate per annum, and n is number of years

So, basically, we need to find out the following:

20,000*((1+10/100)^20) + 20,000*((1+10/100)^19) + 20,000*((1+10/100)^18) + .. + 20,000*((1+10/100)^1)

= 20,000*[x^20 + x^19 + .. + x], where x = 1 + 10/100

Getting x^n + x^(n-1) + x^(n-2) + … + x is the tricky part.

Say y = x^n + x^(n-1) + x^(n-2) + … + x --- (1)

From 1,

y - x^n = x^(n-1) + x^(n-2) + … + x

=> y - x^n + 1 = x^(n-1) + x^(n-2) + … + x + 1---- (2)

Also, from 1,

y = x(x^(n-1) + x^(n-2) + … + x + 1) --- (3)

Combining 2 and 3,

y = x ( y - x^n + 1)

=> y = xy - x^(n+1) + x

=> x^(n+1) - x = xy - y

=> y(x-1) = x(x^(n) - 1)

=> y = x(x^(n) - 1) / (x - 1)

check if formula is correct (example):

What is 3^4 + 3^3 + 3^2 + 3 ?

= 81 + 27 + 9 + 3
= 120

Using the formula (y = x(x^(n) - 1) / (x - 1)):

3^4 + 3^3 + 3^2 + 3 = 3(3^4 - 1) / (3 -1)

= 3(81 - 1) / (3 - 1)
= 3*80/2
= 3*40
= 120

Now, using the formula, after 20 years, the total fund value would be 20,000 * [1.1(1.1^20 - 1)/(1.1-1)]; x = 1 + 10/100 = 110/100 = 1.1

= 20,000 * [1.1 * 6.7275 / 0.1]
= 20,000 * 11 * 6.7275
= $1,480,050

Hope you find it useful.

Regards,
vishal

The math logic is perfect. A calculatio error in the penultimate step. Missed out on (1.1^20 -1) part. The correctanswer is $126005/-

Thanks, Vishal - just want to check something. Do you mean that you created this problem yourself or you found it somewhere? If you found it somewhere, please cite the source of the problem - we have to cite anything we do not write ourselves. If you made it yourself, thanks for the contribution!