Sylvia invests her money in an account earning interest based on simple discount at a $2$$\%$ annual rate. What is her effective interest rate in the fifth year?
The formula for effective interest rate is $r = (1+\frac in)^n -1$
The way I would do this is find $i$ which is $0.0204$
Then I would plug everything in $r = (1+ \frac {0.0204}{5})^5-1 $ and my answer is $2.056 \%$
However, when looking up solutions, this is what they did
They took $i$ and multiplied it by $5$ which equals to $0.1020$
Then their effective interest rate formula would look like this $r = (1 + 0.1020)^{1/5}-1$ and they get $.0196 \%$
What am I doing wrong?
The annual effective rate of interest for year $t$, which we denote by $i(t)$, is the ratio of the amount of interest earned in a year, from time $t−1$ to time $t$, to the accumulated amount at the beginning of the year (i.e., at time $t−1$): $$ i(t)=\frac{a(t)-a(t-1)}{a(t-1)} $$ For the simple-discount method, we have $a(t)=\frac{1}{1-dt}$ where $d=2\%$ is the simple discount rate. Observing that $a(5)=\frac{1}{1-0.02\times 5}=\frac{1}{0.9}$ and $a(4)=\frac{1}{1-0.02\times 4}=\frac{1}{0.92}$ $$ i(5)=\frac{a(5)-a(4)}{a(4)}\approx 2.22222\% $$
NOTE: this exercise comes from the book Mathematical Interest Theory, 2nd ed. (Leslie Vaaler,James Daniel): ex. Sec 1.8 n. (2) at pag 65, sol. at pag. 441