Maths Demo (IV)

• S6 Sequences
  • Solve an AS from Two Terms
  • Solve a GS from Two Terms
  • Successive Deposit Problem

S6 Sequences

Solve an AS from Two Terms

In an arithmetic sequence, $T_{3}=9$ and $T_{8}=44$. Find the general term $T_{n}$.

Solution

Formula: $T_{n}=a+(n-1)d$

$\left\{ \begin{aligned} T_{3} &= a+(3-1)d \\ &= 9 \\ T_{8} &= a+(8-1)d \\ &= 44 \end{aligned} \right. \to \left\{ \begin{aligned} a+2d &= 9 \\ a+7d &= 44 \end{aligned} \right.$

∴     $a=-5$ and $d=7$

∴     $T_{n}=-5+(n-1)(7)$

Solve a GS from Two Terms

In a geometric sequence, $T_{4}=54$ and $T_{8}=4374$. Find the general term $T_{n}$.

Solution

Formula: $T_{n}=ar^{n-1}$

$\left\{ \begin{aligned} T_{4} &= ar^{4-1} \\ &= 54 \\ T_{8} &= ar^{8-1} \\ &= 4374 \end{aligned} \right. \to \left\{ \begin{aligned} ar^{3} &= 54 && ...(1) \\ ar^{7} &= 4374 && ...(2) \end{aligned} \right.$

$\begin{aligned} \dfrac{(2)}{(1)}:\ \dfrac{ar^{7}}{ar^{3}} &= \dfrac{4374}{54} \\ r^{4} &= 81 \\ r &= 3 \text{ or } -3 \end{aligned}$

If $r=3$:

Put $r=3$ into (1), we have $a=2$

∴     $T_{n}=2(3^{n-1})$

If $r=-3$:

Put $r=-3$ into (1), we have $a=-2$

∴     $T_{n}=-2(-3)^{n-1}$

Successive Deposit Problem

At the beginning of each year, Alan deposits $200 into the bank at 10% p.a. compounded yearly.

Let $A_{n}$ be the amount at the end of the nth year. Express $A_{n}$ in terms of $n$.

Solution

Study the pattern:

$\begin{aligned} A_{1} &= 200(1.1) \\ A_{2} &= (A_{1}+200)(1.1) \\ &= 200(1.1)^{2}+200(1.1) \\ A_{3} &= (A_{2}+200)(1.1) \\ &= 200(1.1)^{3}+200(1.1)^{2}+200(1.1) \\ A_{4} &= (A_{3}+200)(1.1) \\ &= 200(1.1)^{4}+200(1.1)^{3}+200(1.1)^{2}+200(1.1) \\ &\vdots \end{aligned}$

Following the pattern, we have:

$\begin{aligned} A_{n} &= 200(1.1)^{n}+200(1.1)^{n-1}+200(1.1)^{n-2}+\ldots+200(1.1)^{2}+200(1.1) \\ &= 200(1.1)+200(1.1)^{2}+\ldots+200(1.1)^{n-2}+200(1.1)^{n-1}+200(1.1)^{n} && \color{red}{\leftarrow \text{ reverse the sequence}} \\ &= \dfrac{200(1.1)\left[1.1^{n}-1\right]}{1.1-1} && \color{red}{\uparrow \text{ GS with } a=200(1.1) \text{ and } r=1.1} \\ &= \$2200(1.1^{n}-1) \end{aligned}$