A geometric series can be described as series of numbers that differ by a constant ratio or constant multiple.

The following numbers sequence fits a geometric series:

A , AR , AR^2, ..., AR^(n-1), AR^n, ... AR^∞ (an infinite series).

The first term of the series is A and the constant ratio or

**constant multiple is R.**Each successive term is r multiplied by the previous term. The sum of the first n terms of the GP, Sn is given by:

Sn = A + AR + AR^2+ ...+ AR^(n-1) (notice the last term has a power of R of n-1)

To solve for Sn, we will use a trick. First, multiply Sn by R to give Sn*R

Sn*R= AR + Ar^2+ ...+ Ar^n (notice the last term has a power of R of n)

we subtract Sn*R from Sn to give:

Sn - Sn*R=[A + AR + AR^2+ ...+ AR^(n-1)]-[AR + AR^2+ ...+AR^(n-1) +AR^n ] = A-AR^n

Sn(1-R) = A(1- R^n)

**therefore Sn = [A(1- R^n)] / (1-R) or Sn = [A/(1-R)]***

**(1- R^n)**

**(formula for sum of the first n terms)**

A similar analysis can be used to show the sum to infinity of the series, provided the series converges (i.e. terms get smaller, which will be the case if the constant ratio is less than zero).

S∞ = A +AR + AR^2 +...+AR^(n-1)+AR^n +...

S∞*R= AR + AR^2 +...+AR^n+AR^(n +1)+...

S∞-S∞*R = [A +AR + AR^2 +...+AR^(n-1)+AR^n +... ]-[AR + AR^2 +...+AR^n+AR^(n +1)+... ]

S∞-S∞*R= A

**S∞(1-r ) = A or S∞ =A / (1-R)**

**(formula for sum of infinite terms of the GP)**

**These two formulas form the basis for deriving the annuity, growing annuity, perpetuity and growing perpetuity formulas.**

**Let is now apply these formulas to finance. r in this application is a discount rate, and not a constant ratio, R as used above.**.

An annuity is a series of payments of the same amount, C paid at regular intervals that end at period n. Thus the following payments are an annuity,

C,C,C,...,C paid at periods 1,2,3,...,n (could be years 1,2,3,...,n if payments are annual).

The present values of these cash flows is also series payments of the form:

C/(1+r), C/(1+r)^2, C/(1+r)^3,...,C/(1+r)^n

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=1/(1+r)

including these terms in the equation for the first n terms of a GP gives

PV(A) = {[C/(1+r)] / [1 - {1/(1+r)}]}* [1-{1/(1+r)}^n]

from

**Sn = [A/(1-R)]***

**(1- R^n)**

**with some algebraic manipulation the above expression becomes:**

PV(A) = {[C/(1+r)] / [(1 +r - 1)/(1+r)]} * [1-{1/(1+r)}^n]

PV(A) = {[C/(1+r)] / [ r/(1+r)]} * [1-{1/(1+r)}^n]

**PV(A) = [C/r]* [1-{1/(1+r)^n}]**

**the desired result for the present value of an annuity**

**It may be worthwhile to work through the derivation once to verify the algebra.**

For the present value of a growing annuity, the present values of the cash flows is given by:

C/(1+r), C(1+g)/(1+r)^2, C(1+g)^2/(1+r)^3,...,C(1+g)^(n-1)/(1+r)^n

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=(1+g)/(1+r)

PV(GA) = {[C/(1+r)] / [1 - {(1+g)/(1+r)}]}* [1-{(1+g)/(1+r)}^n]

from

**Sn = [A/(1-R)]***

**(1- R^n)**

with some algebraic manipulation the above expression becomes:

PV(GA) = {[C/(1+r)] / [(1 +r -1-g)/(1+r)]} * [1-{(1+g)/(1+r)}^n]

PV(GA) = {[C/(1+r)] / [( r-g)/(1+r)]} * [1-{(1+g)/(1+r)}^n]

**PV(GA) = [C/(r-g)]* [1-{(1+g)/(1+r)}^n]**

**the desired result for the present value of a growing annuity with g<r**

**where g is the growth rate of payments C between each period.**

**The formulas for perpetuities are easier. For a simple perpetuity with no growth, the cash flow sequence is:**

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=1/(1+r)

The sum to infinity of the series is given by:

PV(P) = [C/(1+r)] / [ 1 -{1/(1+r)}]

from

**S∞ =A / (1-R)**

with some algebraic manipulation the above expression becomes:

PV(P) = [C/(1+r)] / [ (1+r -1)/(1+r)] = [C/(1+r)] / [ r/ (1+r)] = C/r

**PV(P) = C/r which is the desired result.**

For a growing perpetuity, the cash flow sequence is:

C/(1+r), C(1+g)/(1+r)^2, C(1+g)^2/(1+r)^3,...,C(1+g)^(n-1)/(1+r)^n, ...

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=(1+g)/(1+r)

PV(GP) = [C/(1+r)] / [ 1 -{(1+g)/(1+r)}]

PV(GP) = [C/(1+r)] / [ (1+r -1-g)/(1+r)] = [C/(1+r)] / [(r-g)/ (1+r)] = C/(r-g)

**PV(GP) = C/(r -g) which is the desired result.**

**Summary:**

**Sn = [A/(1-R)]*(1- R^n) - Sum of the first n terms of a geometric progression**

**S∞ =A/(1-R) -**

**Sum of the infinite terms of a geometric progression**

**PV(A) = [C/r]* [1-{1/(1+r)^n}] - Present value of an annuity**

**PV(GA) = [C/(r-g)]* [1-{(1+g)/(1+r)}^n] - Present value of a growing annuity**

**PV(P) = C/r**

**- Present value of a perpetuity**

**PV(GP) = C/(r -g)**

**- Present value of a growing perpetuity**

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