§-Quick questions

NSWMaths AdvancedYear 12: Financial Mathematics

Quick questions on Geometric sequences and series for HSC Maths Advanced financial modelling

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are geometric sequences?

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A geometric sequence has a constant ratio

r

between consecutive terms. With first term

a

What is limiting sum (infinite series)?

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|r| < 1

, then

r^n \to 0

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

What is compound interest as a geometric sequence?

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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

What is off-by-one on the exponent?

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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

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.

What is sum starting at the wrong term?

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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.

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