Applications of GP in Business: Compound Interest and Annuities

Compound Interest as a Geometric Progression

Under compound interest, the amount at the end of each year is obtained by multiplying the previous year's amount by a fixed factor $(1+i)$, where $i$ is the rate of interest per annum (as a decimal). This means the year-end amounts themselves form a GP:

with first term $A_0$ (the principal) and common ratio $(1+i)$. The amount after $n$ years is simply the $(n+1)$th term of this GP:

— which is exactly the compound-interest formula met in Accountancy/Business Mathematics, now understood as a direct application of the GP nth-term rule ($T_{n+1} = ar^{n}$ with $a = A_0$, $r = (1+i)$).

The same idea, run in reverse (a fixed factor less than 1 multiplying the previous year's value) models depreciation of an asset by a fixed percentage each year: if a machine worth $V_0$ depreciates at rate $d$ per annum, its value after $n$ years is $V_n = V_0(1-d)^n$ — again a GP, with common ratio $(1-d)$.

Annuities: Future Value Using the GP Sum Formula

An annuity is a series of equal payments (instalments) made at regular intervals. For an ordinary annuity of $R$ per period, invested at interest rate $i$ per period for $n$ periods (each payment made at the end of its period), the future value (the total accumulated value of all payments, with interest, at the end of the last period) is found by treating the accumulated value of each payment as a term of a GP and summing them:

$FV = R + R(1+i) + R(1+i)^2 + \cdots + R(1+i)^{n-1} = R \cdot \frac{(1+i)^n - 1}{i}$ …