Use the formulas for the nth term and the sum of n terms of a geometric sequence, and the limiting sum, in financial contexts

What this dot point is asking

NESA wants you to recognise geometric sequences and series, apply the formulas for the $n$th term, the sum of $n$ terms and the limiting sum, and use them in financial contexts such as repeated payments, depreciation, and perpetuities.

The answer

Geometric sequences are the engine behind every formula in this topic. The moment a quantity is repeatedly multiplied by the same factor each period, a fixed interest rate, a fixed depreciation rate, a stream of equal payments each discounted by one more period, you have a geometric sequence, and its sum is a geometric series. Compound interest, declining-balance depreciation, the future value of an annuity and the present value of a perpetuity are all the same three formulas wearing different clothes. The one genuinely new idea is the limiting sum: when the multiplier has size less than one, an infinite list of terms can still add to a finite total.

Geometric sequences

A geometric sequence has a constant ratio $r$ between consecutive terms. With first term $a$,

So $T_1 = a$, $T_2 = a r$, $T_3 = a r^2$, and so on. The defining test is that dividing any term by the one before it always gives the same number $r$ (compare an arithmetic sequence, where you subtract to get a constant difference).

Geometric series (finite sum)

The sum of the first $n$ terms is

Both forms are equivalent. Use whichever keeps the numerator positive in your particular case.

Limiting sum (infinite series)

If $|r| < 1$, then $r^n \to 0$ as $n \to \infty$, so $S_n = \dfrac{a(1 - r^n)}{1 - r} \to \dfrac{a(1 - 0)}{1 - r}$, and the series converges to

If $|r| \ge 1$, the terms do not shrink to zero, the partial sums grow without bound (or oscillate), and the limiting sum does not exist.

Why an infinite sum can be finite, stage by stage

The limiting sum feels paradoxical until you watch it. Take the series $1 + \tfrac{1}{2} + \tfrac{1}{4} + \cdots$, with $a = 1$ and $r = \tfrac{1}{2}$, so $S_\infty = \dfrac{1}{1 - 1/2} = 2$. Lay each term end to end on a number line: every new term is half the length of the last, so it closes half of whatever gap to $2$ remains. The partial sum keeps moving right but never passes $2$.

Stage 1, the first term. Lay down $1$. The running sum is $S_1 = 1$, exactly halfway to the limit.

Stage 2, add a half. The next term is $\tfrac{1}{2}$, closing half the gap from $1$ to $2$. Now $S_2 = 1.5$.

Stage 3, add a quarter. The third term $\tfrac{1}{4}$ closes half the gap again, reaching $S_3 = 1.75$. Each step leaves exactly half the previous gap.

Stage 4, keep halving the gap. Adding $\tfrac{1}{8}$, then $\tfrac{1}{16}$, and so on, the sum reaches $S_5 = 1.9375$ and edges ever closer to $2$ without reaching it. The remaining gap after $n$ terms is exactly $r^n \cdot \dfrac{a}{1-r}$, which shrinks to zero, so $S_\infty = 2$.

Compound interest as a geometric sequence

A principal $P$ at compound rate $r$ per period produces the sequence of balances $P, P(1 + r), P(1 + r)^2, \dots$ with common ratio $1 + r$. The balance after $n$ periods is the $(n + 1)$th term, which gives the familiar $A = P(1 + r)^n$.

Depreciation

An asset depreciating at rate $d$ per period has values $V_0, V_0(1 - d), V_0(1 - d)^2, \dots$, a geometric sequence with ratio $1 - d$. The value after $n$ periods is $V_n = V_0 (1 - d)^n$. This is the "declining balance" method.

Repeated payments and perpetuities

A series of equal payments made at regular intervals forms a geometric sum once each payment is moved to a common time using the compound interest factor. If the payments stop after $n$ terms, use the finite sum $S_n$ (this is the future-value-of-annuity formula). If they continue forever and the per-period discount factor $v = (1 + r)^{-1}$ satisfies $|v| < 1$ (which it always does for $r > 0$), the limiting sum gives the present value of a perpetuity: $\text{PV} = \dfrac{\text{payment}}{r}$.

How exam questions ask about geometric sequences and series

The context is dressed up, but each version reduces to identifying $a$ and $r$ and choosing the right formula:

• "Find the value after $n$ years" of an asset that loses a fixed percentage each year. Declining-balance depreciation: $V_n = V_0(1 - d)^n$, a geometric sequence with ratio $1 - d$.
• "Find the $n$th term" or "which term equals $X$?" Use $T_n = ar^{n-1}$; for "which term", set it equal to $X$ and solve for $n$ with logs.
• "Find the sum of the first $n$ terms" or a total of repeated equal deposits. Finite sum $S_n = \dfrac{a(r^n - 1)}{r - 1}$.
• "Find the limiting sum" or "explain why a limiting sum exists". State $|r| < 1$ first, then $S_\infty = \dfrac{a}{1 - r}$.
• "Value a scholarship / pension that pays $X forever." A perpetuity: present value $= \dfrac{X}{r}$ (the limiting sum of the discounted payments).
• "Show that the balance / total is a geometric series." Write the first few terms, state $a$ and $r$, then apply the sum formula; markers want the structure made explicit.