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

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.

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 $n \to \infty$, and the series converges to

If $|r| \ge 1$, the limiting sum does not exist.

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 after each payment is moved to a common time using the compound interest factor. When the payments continue forever and the discount rate satisfies $|v| < 1$, the limiting sum gives the present value of a perpetuity.

Worked examples

nth term and finite sum

Find $T_8$ and $S_8$ for the geometric sequence $3, 6, 12, 24, \dots$.

$a = 3$, $r = 2$.

$T_8 = 3 \cdot 2^7 = 3 \cdot 128 = 384$.

$S_8 = \frac{3(2^8 - 1)}{2 - 1} = 3 \cdot 255 = 765$.

Depreciation

A car worth \32000$depreciates at$20%$per year. Its value after$5$ years is

V = 32000 (0.8)^5 = 32000 \cdot 0.32768 \approx \10485.76$.

Limiting sum

Find the limiting sum of $8 - 4 + 2 - 1 + \cdots$.

$a = 8$, $r = -\frac{1}{2}$, $|r| = \frac{1}{2} < 1$.

$S_\infty = \frac{8}{1 - (-1/2)} = \frac{8}{3/2} = \frac{16}{3}$.

Time to repay using a geometric sum

If you deposit \100$at the end of each year into an account paying$5%$per annum, the balance just after the$n$th deposit is

The \100$deposited today has had no compounding yet; the deposit made$n - 1$years ago has compounded$n - 1$ times. This is the future-value-of-annuity setup, developed in the annuities dot point.

Perpetuity

A scholarship pays \5000$at the end of each year forever, discounted at$4%$per annum. The present value is the limiting sum of$5000 v + 5000 v^2 + \cdots$with$v = \frac{1}{1.04}$.

\text{PV} = \frac{5000 v}{1 - v} = \frac{5000 / 1.04}{1 - 1/1.04} = \frac{5000}{1.04 - 1} = \frac{5000}{0.04} = \125000$.

A perpetuity is the payment divided by the per-period rate.

Common traps

Wrong choice of $a$. The first term $a$ is the first term of the sequence as written. In $T_n = a r^{n - 1}$, $a = T_1$, not $T_0$.

Off-by-one on the exponent. $T_n$ uses $r^{n - 1}$, not $r^n$. The sum $S_n$ uses $r^n$. The $n$th term of $P, P(1 + r), P(1 + r)^2, \dots$ is $P(1 + r)^{n - 1}$, but the compound interest balance after $n$ periods is $P(1 + r)^n$ because we count compounding events, not list positions.

Forgetting the convergence condition. The limiting sum formula needs $|r| < 1$ strictly. For $r = 1$ the sequence is constant and the partial sums diverge. For $|r| \ge 1$ but $r \neq 1$, the partial sums grow without bound or oscillate.

Confusing depreciation rate with multiplier. A $15\%$ depreciation per year means a multiplier of $0.85$ per year, not $0.15$. The new value is $0.85$ times the old.

Sum starting at the wrong term. Some questions list payments starting one period from now (an "ordinary annuity"), others start immediately (an "annuity due"). The first term and the number of compounded periods change accordingly.

In one sentence

A geometric sequence has $T_n = a r^{n - 1}$ with finite sum $S_n = \frac{a(1 - r^n)}{1 - r}$ and limiting sum $\frac{a}{1 - r}$ when $|r| < 1$, and these formulas underlie compound interest, depreciation, annuities and perpetuities.