**Topic**

A person deposits amount A (VND) in a bank. The bank’s interest rate schedule is as follows:

Divide each year into m terms and interest rate r%/year. Know that if you do not withdraw money from the bank, after each term, the amount of interest will be entered into the original capital. Prove that the amount received (including capital and interest) after n (years) deposit is \({S_n} = A. {\left( {1 + \frac{r}{{100m}}} \right) ^{mn}}\) (VND), if within this period the depositor does not withdraw money and the interest rate does not change.

**Detailed explanation**

We prove: “The amount received (including capital and interest) after p (term) deposit is \({T_p} = A. {\left( {1 + \frac{r}{{100m}} ) } \right)^p}\) (bronze).”

Thus, because each year has m terms, after n years is respectively mn terms, from which we deduce the amount received (including capital and interest) after n (years) deposit is \({S_n} = {T_{mn}} = A. {\left( {1 + \frac{r}{{100m}}} \right)^{mn}}\) (bronze), which must be proved.

+ Prove: “The amount received (including capital and interest) after p (term) deposit is \({T_p} = A. {\left( {1 + \frac{r}{{100m}} ) } \right)^p}\) (bronze).”

Step 1: When \(p = 1\) we have

The interest rate for m terms (or 1 year) is r% => The interest rate for each term is \(\frac{{r\% }}{m} = \frac{r}{{100m}}\)

The amount received (including capital and interest) after 1 (term) deposit is: \(A + A.\frac{r}{{100m}} = A{\left( {1 + \frac{r) }{{100m}}} \right)^1} = {T_1}\) (bronze)

So the statement is true for \(p = 1\)

Step 2: With k being an arbitrary positive integer whose proposition is true, we have to prove the proposition to be true for k + 1, that is:

“The amount received (including capital and interest) after \(k + 1\) (term) deposit is \({T_{k + 1}} = A. {\left( {1 + \frac{) r}{{100m}}} \right)^{k + 1}}\) (bronze).”

Indeed, by the assumption of induction we have:

“The amount received (including capital and interest) after k (term) deposit is \({T_k} = A. {\left( {1 + \frac{r}{{100m}}} \right) ^k}\) (dong).”

=> The amount received (including capital and interest) after \(k + 1\) (term) deposit is:

\(\begin{array}{l}A. {\left( {1 + \frac{r}{{100m}}} \right)^k} + A. {\left( {1 + \frac{r) }{{100m}}} \right)^k}.\frac{r}{{100m}}\\ = A. {\left( {1 + \frac{r}{{100m}}} \right) ^k}\left( {1 + \frac{r}{{100m}}} \right)\\ = A. {\left( {1 + \frac{r}{{100m}}} \right)^ {k + 1}} = {T_{k + 1}}\end{array}\)

So the statement is true for k+1. Thus, according to the principle of mathematical induction, the statement is true for all \(p \in \mathbb{N}*\).

Thus, after n years (meaning term mn), the amount received (including capital and interest) is \({S_n} = A. {\left( {1 + \frac{r}{{100m) }}} \right)^{mn}}\)(bronze).