SL 1.4—Financial apps – compound interest, annual depreciation Questionbank

Write the future values $V_A,V_B$ and simplify $V_B$.

Prove that $V_A>V_B$ for $n\ge2$ using $(1+r)^n>1+nr$.

Show that the percentage shortfall $\dfrac{V_A-V_B}{V_A}$ equals $1-\dfrac{(1+r)^n-1}{nr(1+r)^n}$ and decreases as $r$ decreases or $n$ increases (qualitative).

A lump sum $P$ is invested today. Write the value after $n$ years.

A target $k>1$ is set so that the balance must reach $kP$. Find the least integer $n$ required, using logarithms.

Two different rates $r_1,r_2>0$ are offered for the same horizon $n$. Prove using log/exponent laws that $P(1+r_1)^n>P(1+r_2)^n$ iff $r_1>r_2$.

If $g>0$, find $n$ so that $R_n=2R_0$.

If $g<0$ (decline), show that the half-life $H$ (time until $R_H=\tfrac12 R_0$) equals $H=\dfrac{\log(\tfrac12)}{\log(1+g)}$. Explain why $H>0$.

Prove the law $\log_a(xy)=\log_a x+\log_a y$ from the exponential identity $a^{u+v}=a^u a^v$ using part 1’s reasoning structure.

Find the largest integer $n$ satisfying the warranty.

If the firm upgrades to a two-stage schedule: first $k$ years at rate $d_1$, then at $d_2$ ($0<d_1,d_2<1$). Show
$V_n=\begin{cases} V_0(1-d_1)^n,& n\le k,\\ V_0(1-d_1)^k(1-d_2)^{\,n-k},& n>k. \end{cases}$

Prove $\log_a\!\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$ and use it to linearize the inequality for the warranty in the two-stage case.

Show that, at the end of 5 years, the machine will be worth approximately $16477, correct to the nearest dollar.

Another company also purchases a machine for $25000. At the end of 3 years, this machine is worth $18000. Find the yearly depreciation rate for this machine.

An investor places a lump sum $P$ at the start of month 1. Find the value after $n$ months.

Another investor pays a fixed amount $M$ at the end of each month for $n$ months at the same rate $i$. Using the geometric-series sum, write the future value after $n$ months in closed form.

Suppose the same total principal $P$ is contributed over the $n$ months by splitting it equally: $M=\dfrac{P}{n}$ paid at the end of each month. Prove that for all integers $n\ge 2$, the lump-sum strategy in part 1 yields a strictly larger future value than this split-payment strategy in part 2 (You may use Bernoulli’s inequality: for $x>0$ and integer $n\ge 2$, $(1+x)^n>1+nx$.)

A target multiplier $k>1$ is set so that the lump sum in (a) must reach at least $kP$. Find the least integer $n$ satisfying the requirement, expressing your answer using logarithms (no numerical evaluation needed).