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 n1 (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 n1 (3) 3. Notice that all terms on the right hand side except for the first and last term, a, aR n1 , are cancelled. So, equation (3) becomes,

(4)
S n (1 R) a aR n1
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 n1 )
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

...
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