Investment Projects & Firms’ Valuation
Fernando Martínez, MBA, CRM
MMXXII
IV. Annuities & Perpetuities
Fernando Jesús Martínez Eissa, MBA
MMXXII
School of Economic and Business Sciences
An
arithmetic sequence
, is a series of terms which difference between
two consecutive terms is a constant, for example:
1.
6,11,16,21,26,31,36,41. The constant difference (
d
) is: 5
2.
54,50,46,42,38,24,30,26,22,18. The constant difference
d
=4
Arithmetic Sequences
To build a 5term sequence considering
a
as its first term and
d
, its
difference, then the sequence shall be:
d
a
d
a
d
a
d
a
a
4
,
3
,
2
,
,
+
+
+
+
We can observe that for an “
n
” terms sequence, the last term could be
written as:
࠵? = ࠵? +
࠵? − 1 ࠵?
The sum of an arithmetic sequence is:
)
*
+,*
./0 1
࠵? =
࠵?
2࠵? +
࠵? − 1 ࠵?
2
=
࠵?
2
࠵? + ࠵?
Arithmetic Sequences
Example 1
Find 15
th
term and the addition of the first 10 terms of the arithmetic
sequence : 6, 11, 16, 21,…
Solution
We have:
a
=6,
d
=5 and
n
=15. Also we know:
(
)
(
)
285
51
6
5
2
=
+
=
+
=
u
a
n
s
To add the first 10 terms we need to find the 10
th
term, applying the as
we do before:
a
=6,
d
=5 and
n
=10, therefore
n
=51.
Therefore the 15
th
term is:
Then, the sum of the first 10 terms is:
࠵? = ࠵? +
࠵? − 1 ࠵?
࠵? = 6 +
15 − 1 5
࠵? = 76
Arithmetic Sequences
Example 2
Juan borrows $4,000 and agrees to pay $400 each month at 2.5% simple
interest rate. Find the total amount of interest to be repaid.
Solution
Mes
Pagos
Saldo Insoluto
Intereses
4,000
100
1
400
3,600
90
2
400
3,200
80
3
400
2,800
70
4
400
2,400
60
5
400
2,000
50
6
400
1,600
40
7
400
1,200
30
8
400
800
20
9
400
400
10
10
400
0
0
550
The
amount
of
interest
paid
is:
(4,000)*2.5%=100;
(3,600)*2.5%=90;
(3,200)*2.5%=80.
It is also known that Juan will pay 10
installments (4,000/400=10).
Therefore, the sum of interest paid is:
(
)
[
]
[
]
550
90
200
5
1
2
2
=

=

+
=
d
n
a
n
s
Arithmetic Sequences
A
geometric sequence
, is a series of terms that can be obtained
multiplying the previous term by one fix number called a rate, for
example:
1.
4,8,16,32,64,128,256,512. The rate (
r
) is: 2
2.
729,486,324,216,144,96,64. The rate (
r
) is:
r
=2/3
8
7
6
5
4
3
2
,
,
,
,
,
,
,
,
ar
ar
ar
ar
ar
ar
ar
ar
a
To build a 9term geometric sequence, considering “
a
”
as the first term
and “
r
”
as the increasing rate, we have:
We can observe that for an “
nterms
” the last term could be expressed
as:
(
)
( )
i
ar
u
n
1

=
Geometric Sequences
Let
s
be the sum of the geometric sequence:
Then,
1
3
2
,
,
,
,
,

n
ar
ar
ar
ar
a
!