If you invest $\$10$ at beginning of each month for $10$ years, how much will you have at end of the $10$ years

economicsfinancegeometric seriessequences-and-seriessummation

Problem

If you invest a dollar at “$6\%$ interest compounded monthly,” it amounts to $(1.005)^n$ dollars after $n$ months. If you invest $\$10$ at the beginning of each month for $10$ years ($120$ months), how much will you have at the end of the $10$ years?

[Source : Mary Boas Mathematical Method in the Physical Sciences $3$Ed: $1.1.13$]

My Understanding

It's a monthly compound interest problem where you start with $\$10$ with a interest rate of $6\%$ and every month you invest an additional $\$10$. The question asks to find the final amount after $10$ years of investing ($120$ months).

I am confused after looking at the solution manual

It takes the sum of the series
$$S=10(1.005)+10(1.005)^2+10(1.005)^3+…10(1.005)^{120}$$ and uses the partial sum formula for the geometric series to calculate the sum as $1646.99$

When I was solving the problem, my series wasn't geometric:
$$S=10(1.005) + (10(1.005)+10)1.005 + ((10(1.005)+10)1.005)1.005+…$$
which simplifies to
$$10(1.005+(1.005^2+1.005)+(1.005^3+1.005^2+1.005)+…)$$

I have two questions.

  1. Why it is the case the solution represents this problem with a geometric series if $a$, the initial amount, alters due to the additional $\$10$ investment each month.
  2. Is there a formula to calculating the sum of the series I wrote?

Thanks!

Best Answer

For part (a), I think that you are looking at it from the wrong perspective. Suppose that instead of making monthly deposits into one account, each monthly deposit went into a separate bank account. So, you end up with $(120)$ separate bank accounts. Then, the issue is, what is the total value of all of these accounts, after $(10)$ years.

If an account is opened with $(k)$ investment periods remaining, with $k \in \{1,2,\cdots, 120\}$, then after 10 years, that specific account will have grown to $10(1.005)^k$. This explains the offered solution.

As far as your 2nd question:

  • It is difficult to decipher, because you didn't use mathJax to format the math.

  • I am unable to understand what analysis you used to conjure the math expression that corresponds to your 2nd question.

  • If, after reading this answer, you are able to edit your original posting with MathJax, and you then (still) want to try to find a nice closed form expression that represents the math formula re your 2nd question, please leave a comment following my answer. Then, I will take a crack at it.

Addendum
A more formal (i.e. less intuitive) attack on part (a) may be done via induction.

After $(1)$ month, your total balance is

$\displaystyle (10)[(1.005)^1 + 1].$

Suppose that after $(K)$ months, your total balance is

$\displaystyle(10)\left[\sum_{i=0}^{K} (1.005)^i\right].$

Then, in month $(K+1)$ your balance will become

$\displaystyle (10) \times \left\{\left[(1.005) \times \sum_{i=0}^{K} (1.005)^i\right] + 1\right\} = (10) \times \left[\sum_{i=0}^{K+1} (1.005)^i\right] .$

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