This document presents information about geometric sequences and series. It defines geometric sequences as sequences where each term is found by multiplying the previous term by a common ratio. Geometric series are the sums of the terms in a geometric sequence. Formulas are provided for calculating individual terms, the common ratio, and the sum of finite geometric series. Applications of geometric series discussed include calculating interest, average increases, net present values, and loan repayments. Several example problems demonstrate using the formulas to find terms, common ratios, and sums.

1. Presentation
Title:
Applications of Geometric series in Business.
Prepared By:
Wasif Ali Syed
(18221554-001)

2. Sequence
A Sequence is a set of things (usually numbers) that
are in order.

3. Series
The sum of an indicated number of terms in a sequence
is called a series. e.g., the sum of the first seven terms
of the sequence is the series.
4+8+12+16+20+24+28

4. Geometric Sequences
A sequence {an}n=0 = {a0, a1 ,a2 ,a3 , …}is an ordered set
of numbers. The index of each term of the sequence
indicates the position or order in which specific data is
found. This order is very important. For example, the
sequence {1,3,5,7,9, …} differs from the sequence
{9,7,5,3,1, …} , even if the terms are the same.

5. Geometric series
Given {an} = {a0, a1r ,a2r2,a3r3, …}, a geometric sequence
of common ratio r. A geometric series is the sum of the
elements of a geometric sequence a0 + a1r + a2r2 + a3r3
+….
A series can be finite (with a finite number of terms) or
infinite.

6. Formula to evaluate n term in geometric
sequence
an = a1rn – 1
Where
a1 is the first term
r is the common ratio
n is the number of terms

7. Formula to evaluate common ratio in
geometric sequence
Where
r is the common ratio
a2 is the 2nd term.
a1 is the 1st term.
r
a
a
 2
1

8. Formula to evaluate a geometric series
Where
Sn Sum of GP with n term
a1 first term
n number of terms
r common ratio

9. Application of Geometric Series
1. It is used in the calculation of interest.
2. It is used to find the average increase in sale,
production, or other economic or business data.
3. It is theoretically considered to be the best average
in the construction of index number.

10. Application of Geometric Series
4. It is used in accountancy in finding the Net Present
Value of projects.
5. It is used in calculation of repayments of loans and
values of investments.

11. Question #1
Write the first five terms of the geometric sequence
whose first term is a1 = 9 and r = (1/3).
Solution
an = a1rn – 1
a2 = (9)(1/3)2 – 1 = 3
an = 9,3,1,1/3,1/9
a3 = (9)(1/3)3 – 1 = 1
a4 = (9)(1/3)4 – 1 = 1/3
a5 = (9)(1/3)5 – 1 = 1/9

12. Question #2
Find the 15th term of the geometric sequence whose
first term is 20 and whose common ratio is 1.05
Solution
an = a1rn – 1
a15 = (20)(1.05)15 – 1
a15 = 39.599

13. Question #3
Find a formula for the nth term 5, 15, 45, …
Solution
What is the 9th term?
an = a1rn – 1
an = 5(3)n – 1
an = 5(3)n – 1
a9 = 5(3)8
a9 = 32805

14. Question #4
Find the sum of the first 12 terms of the series 4(0.3)n =
4(0.3)1 + 4(0.3)2 + 4(0.3)3 + … + 4(0.3)12
3
.
0
1
)
12
3
.
0
(
2
.
1
2
.
1
12 


S
= 1.714
r
r
a
a
S
n
n



1
1
1

15. Question #5
Find the sum of the first 5 terms of the series 5/3 + 5 +
15 + …
r = 5/(5/3) = 3
r
r
a
a
S
n
n



1
1
1
3
1
)
5
3
(
3
5
3
5
5 


S
= 605/3

16. Thank you