 # Recursion example: Bank account

Suppose you have a bank account with a € 100.- initial deposit which once a year is compounded with $$\gamma = 3\%$$ interest. In this case your gain is $$0.03*100 = 3$$, which means that according to the equation

$S_{t+1}=S_t+gain$

your bank account will hold € 103.- after the first year.

$103 = 100 + 0.03 * 100$

After the second year your $$gain$$ again will be $$\gamma*S_t$$ with $$\gamma$$ still being 0.03 but $$S_t$$ now grown to 103. Hence

$S_{t+2}= 103 + 0.03 * 103 = 106.09$

After the third year your bank account will hold

$S_{t+3}=106.09 + 0.03 * 106.09 = 109.2727$

And in general your account will grow according to the difference equation.

$S_{t+1}=S_t + \gamma * S_t \text{ respectively } S_{t+1}=S_t * (1+ \gamma)$

With constant $$\gamma$$ hence, starting from an initial deposit of $$S_0$$, your account will grow in t time steps according to

$S_t=S_0+\gamma*S_0+\gamma*S_1+\gamma*S_2+...+\gamma*S_t$

or short

$S_t=S_0*(1+\gamma)^t$

According to this formula, after 10 years of annually compounding 3% interest your bank account should hold (rounded to two decimal places):

$S_{10}=S_0*(1+\gamma)^{10}=100*(1+0.03)^{10} = 134.39 \text{Euro}$