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NSWMaths AdvancedQuick questions

Year 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 rr between consecutive terms. With first term aa,
What is limiting sum (infinite series)?
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If ∣r∣<1|r| < 1, then rn→0r^n \to 0 as n→∞n \to \infty, so Sn=a(1−rn)1−r→a(1−0)1−rS_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 PP at compound rate rr per period produces the sequence of balances P,P(1+r),P(1+r)2,…P, P(1 + r), P(1 + r)^2, \dots with common ratio 1+r1 + r. The balance after nn periods is the (n+1)(n + 1)th term, which gives the familiar A=P(1+r)nA = P(1 + r)^n.
What is off-by-one on the exponent?
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TnT_n uses rn−1r^{n - 1}, not rnr^n. The sum SnS_n uses rnr^n. The nnth term of P,P(1+r),P(1+r)2,…P, P(1 + r), P(1 + r)^2, \dots is P(1+r)n−1P(1 + r)^{n - 1}, but the compound interest balance after nn periods is P(1+r)nP(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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