SL 1.4—Financial apps – compound interest, annual depreciation

For compound interest compounded annually, the formula is $A=P(1+r{)}^t$, where $P$ is the principal, $r$ is the annual rate as a decimal, and $t$ is the number of years.

Here $P=2000$ and $r=3\%=0.03$. Substituting: $A=2000(1+0.03{)}^t=2000(1.03{)}^t$.

The expression $A=2000(1.03{)}^t$ correctly models annual compound growth at $3\%$.

We need the correct formula for compound interest with $P=2000$, $r=3\%$ annually.

The multiplier $0.97=1-0.03$ models depreciation or decay at $3\%$, not growth. This is incorrect.

This is simple interest: it adds a fixed dollar amount each year. Compound interest multiplies the balance by a factor each year.

Using $1.3$ as the multiplier confuses $3\%=0.03$ with $30\%=0.30$. The correct decimal for $3\%$ is $0.03$, giving multiplier $1.03$.

$A=2000(1.03{)}^t$ is the only option that correctly applies the compound interest formula.

$P=4500$, $r=0.028$, $n=7$.

$A=4500(1.028{)}^7$

$(1.028{)}^7\approx 1.21389$, so $A=4500\times 1.21389\approx 5462.5$. Wait — let us recompute carefully: $(1.028{)}^7$. $1.028^2=1.056784$, $1.028^4=1.056784^2\approx 1.116793$, $1.028^7=1.028^4\times 1.028^2\times 1.028\approx 1.116793\times 1.056784\times 1.028\approx 1.196113$. Then $A=4500\times 1.196113\approx 5382.51$.

The value after 7 years is approximately \5382.51$.

We must compute $A=4500(1.028{)}^7$ and match it to one of the options.

This is suspiciously round and would imply a growth factor of $\approx 1.21$, which would correspond to a higher rate or more years. It is too large for $2.8\%$ over 7 years.

This value is slightly too low, possibly arising from using $r=0.028$ but making a rounding error in an intermediate step.

This is very close but results from a slight over-rounding of $(1.028{)}^7$. The precise calculation gives \5382.51$.

The correct calculation yields A = 4500(1.028)^7 \approx \5382.51$.