13 Red, Green, Blue
5.0 Reflection and Data Validation 5.1 Introduction
The nth term of a geometric progression and the sum of geometric series is found applicable to a number of areas in economics, finance and business. Among the common application of GP are compound interest, present value, growth rate estimations etc.
Examples
1. A student borrows 600 at 7% interest compounded annually. He pays off the loan at end of 3 years. How much does he pay?
Solution
7% of N 600 = N42
2nd term of the GP is N600 + N42 = N642 Hence the GP is 600 + 642+….
a = 600 r =
07 . 600 1 642=
At the end of the 3rd year the GP will have 4 terms;
4th term = ar3 600 (1.07)3
= 600 (1.2250)
= N735 ∴He pays back N735
2. Olawale saved N30, 000 in the first year of a new job. In each subsequent year, he saved 10% more than in the previous year.
a. How much in total had he saved in 5 years?
b. How many years did he take to save a total of more than N330, 000?
Solution
a. a = N30,000 r = 1 +
1 . 100 1
10 =
Sn =
1 . 0 ) 1 1 . 1 ( 30000 1
1 . 1
) 1 1 . 1 ( 30000
÷
−
− =
−
n
= 300000 (1.1n – 1) Pnt = 5
300000 (1.15 -1) = 300000 (1.611-1) =300000 x 0.611
= N183, 300
b. If he saved a total of N330, 000 after n years.
330, 000 = 300 000 (1.1n – 1) Divide both sides by 300,000.
000 , 300
000 , 300 000 , 300
000 ,
330 =
(1.1n – 1) 1.1 = 1.1n -1
1.1 + 1 = 1.1n 2.1 = 1.1n
Taking the logarithm of both sides Log 2.1 = n log 1.1
n = log1.1 1 . 2 log
= 7.8
... We need a whole number of years hence after 8 years; he has saved more than N330, 000.
3. A man deposit 10,000 at 8% per annum. Find the compound amount at the end of 10 years if (i) Interest is compound annually
(ii) Quarterly (iii) Monthly Solution
i. P = 10,000 = a Sn= a (1 + r)n
S10 = 10,000 [1 + 8%]10
= 10,000 [1 + 0.08]10 = 10,000 [2.159]
= N21, 589
ii. P = 10,000
% 2 4% 8 =
= L
n = 10 x 4 = 40 quarters
Note: There are four quarters in a year, so the interest rate is divided by 4 and the number terms (n) is multiple by 4:
Sn= a(1 + r)n
S40 = 10,000 (1 + 2%)
= 10,000 (1+0.02)40
= 10.000 (1.02)40
= 10,000 (2.208)
= N22,080.
iii. P = 10,000 = a i =
0067 . 100 0
64 .
% 0 67 . 0 12%
8 = = =
n = 12 x 10 = 120
Note: There are twelve months in a year, so the interest rate is divided by 12, charged one a period of 10 years, which is equivalent to 120 months.
S120 = 10,000 [1+ 0.0067]
= 10,000 [0.067]120 = 10.000 [2.228]
= N22, 280
3. Find the present value of N722 receivable in 5 years if the money is worth 12% per annum compounded quarterly.
Solution
( )
p n p[ ]
r nPV = + −
= + 1
1 1
n = 5 X 4 = 20 quarters PV = 722 [1 + 0.12]-20 = 722 [1.12]-20 = 722 [0.5536]
= N4000
4. The population census in 1960 was 95 million. Ten years later the census gave a total population of 115million. Find the annual growth rate.
Solution
Base year t = 0 and P0 = 95, ert, = 1 Then S0 = 95 = Per(0) =P
After ten years
S10 = 115 = Pert =Per(10) S10 = 115 = 95e10r 115 = 95e1or Divide both sides by 95
95 95 95
115 er10
= 1.21 = e10r
Finding the exponential log of both sides In 1.21 = 10r
0.191 = 10r r = 10
191 . 0
r = 0.019 = 1.9%
The annual growth rate is 1.9%
SELF-ASSESSEMENT EXERCISE
i. Bolarinwa starts a job at an annual salary of N800, 000. At the end of each year, his salary increases by 15%.
a. Find his salary during his fourth year in the job.
b. What is the total amount earned in 5 years?
c. After how many years will he have earned a total of N800, 000?
ii. Suppose a dropped ball re-bounces 1/8 of the height when it falls.
How far has it travelled when it reaches the top of the 8th bounce?
iii. Find the compound amount and interest on N 10.000 for 3 years at 8% per annum compounded.
(i) Annually (ii) Quarterly (iii) Monthly.
iv. Find the present value of N 10,000 receivable 5 years from now if money is worth 10% per annum
v. The population of Nigeria in 1997 was estimated at 100 million people. The population is expected to grow at 3.2% every year.
What is the expected population of Nigeria in the year 2015?
4.0 CONCLUSION
Basically, both arithmetic progression (AP) and the geometric progression (GP) have diverse application to economic, business and financial problems. For instance AP is found relevant to income and savings accumulation projection and forecasting while GP is more relevant to compound interest, present value analysis and the growth rates.
5.0 SUMMARY
Just like many other concept in mathematics, sequence and series are of great relevance in solving practical problems. The use nth term of AP and GP as well as their summation or series are useful in estimating future value of sequential and practical oriented problems. Most often problems relating to interest, compounding are easily synthesised into either arithmetic or geometric progression to be able to provide solution to them. It is important to note that no sequence can be arithmetic progression and at the same time be a geometric progression. Hence, the foremost step to know whether the sequence is arithmetic progression or geometric progression after this, it is important to know what the question is interested in testing, while some questions are interested in forecasting future values some are essentially focused on getting the sum of the sequence. The use of appropriate formula is therefore important in solving practical questions involving sequence
6.0 TUTOR-MARKED ASSIGNMENT
1. An employee started on an annual salary of N1m. every year, he received a constant pay rise. After six years, he has earned a total of 8, 250,000. What was the constant pay rise?
2. A contractor for a construction job specifies penalty for delay of completion beyond a certain date as follow: N15, 000 for the first day N16, 000 for the second day etc. How much does a 30 – day delay in completion cost the contractor if the penalty for each day is N1000 more than the previous day?
3. If a person were offered a job N400 the first day, N800 the second day N1600 the third day etc each day wage being double that for the preceding days, how much would he received at the end of 7th days?
4. Find the compound amount and the compound interest on N20, 000 for 5 years at 10% per annum compounded (i) annual (ii) quarterly (iii) monthly.
5. The total enrolment at a state polytechnic is expected to grow at the rate of 10% each year. If the initial enrolment is 120,000
students, what is the expected number of students enrolled at the end of the 5th year?