Coaching Centre
Applications of APs and GPs
Using arithmetic and geometric sequences and series to model salaries, payments and sales, and using limiting sums for totals over an unlimited time.
The formulae
| AP (add $d$) | GP (multiply by $r$) | |
|---|---|---|
| $n$th term | $T_n = a + (n - 1)d$ | $T_n = ar^{n-1}$ |
| Sum of $n$ terms | $S_n = \dfrac{n}{2}(a + l)$ | $S_n = \dfrac{a(r^n - 1)}{r - 1}$ |
| Limiting sum | none | $S_\infty = \dfrac{a}{1 - r}$, for $-1 < r < 1$ |
Examples
A salary starts at $\$60\,000$ and rises by $\$2500$ each year. Find the salary in the 10th year and the total earned over 10 years.
Another salary starts at $\$60\,000$ and rises by $3\%$ each year. Find the salary in the 10th year and the total over 10 years.
A new product sells $10\,000$ units in its first year, and each year's sales are $80\%$ of the year before. What is the most that could ever be sold?
A salary starts at $\$60\,000$ and rises by $\$2500$ each year. Find the salary in the 10th year (digits only).
$60\,000 + 9 \times 2500$.
Find the total earned over those 10 years.
$5(60\,000 + 82\,500)$.
A salary starts at $\$60\,000$ and rises $3\%$ each year. Find the 10th year's salary, to the nearest cent.
$60\,000 \times 1.03^9$.
Sales start at $10\,000$ units and each year are $80\%$ of the year before. Find the total ever sold.
$\dfrac{10\,000}{0.2}$.
A price falls by $20\%$ each year. The common ratio is:
$1 - 0.2$.
Rent rises by $\$20$ per week each year. This is:
A fixed amount is added.
A $\$1000$ prize grows by $5\%$ each year. Find the prize in the 11th year, to the nearest cent.
$1000 \times 1.05^{10}$.
A ball is dropped from $1000$ cm and each bounce reaches $80\%$ of the previous height. Find the height reached after the 3rd bounce.
$1000 \times 0.8^3$.
A limiting sum exists only when:
Then $r^n \to 0$.
New numbers every time. Keep going until you get five in a row.