Sum of Gp

What is the closed-form expression for the below sum of Geometric
Progression (GP) sequence, S n ?
S n  a  aR  aR 2  ...  aR n

(1)

where R is called the common ratio (between consecutive terms) of the GP sequence. The reason why we want to derive a closed-form expression for S n is for the sake of calculating the summation, or otherwise we need to add all terms one-by-one together, which does not make a sense if the number of terms is huge, say a million terms!
Most importantly, we based on the closed-form expression to derive the PV and FV expressions for both ordinary annuity and annuity due.

Steps:
1. Multiply the both sides of equation (1) by the common ratio, R , to have
S n R  aR  aR 2  aR 3  ...  aR n1 (2)
2. Then subtract equation (1) by equation (2), (or vice versa; it doesn’t matter which subtracts which as the result will be the same.), i.e.,
S n  S n R  (a  aR  aR 2 ...  aR n )  aR  aR 2  aR 3  ...  aR n1 (3)
3. Notice that all terms on the right hand side except for the first and last term, a, aR n1 , are cancelled. So, equation (3) becomes,
(4)
S n (1  R)  a  aR n1
4. Remember our objective is to calculate S n . From equation (4), S n is obvious to equal to,

a  aR n 1 a(1  R n1 )
Sn 

(1  R)
(1  R)

(*)

5. We are done. Equation (*) is the closed-form expression of that we want to obtain. Applications:
1. The PV for ordinary annuity, e.g. constant end-of-period cash flows, C, for t periods. PV 

C
C
C

 ... 
2
1  r (1  r )
(1  r ) t

(A1)

Comparing equation (A1) with equation (1), we can see that the PV is a summation of GP sequence, with a  C and the common ratio R 

1
. Therefore,
1 r

the closed-form expression, i.e., also the summation equals to

C
1
1
C
C
C 1 

1 

t t 1
1  r  (1  r ) 
(1  r ) t  C 
1  r (1  r )
1
PV 


 1 

1 r r r  (1  r ) t 
1
1 r
1 r

(A2)

which