ACKNOWLEDGEMENT We would like take this wonderful opportunity to wish a lot of thanks from our bottom of heart. We have learnt many things directly and indirectly throughout this mathematics coursework. All the more, we gained many extra knowledge and it was a great fun of doing such a coursework as it involves something regarding our daily life. We practised to be punctual when a meeting was put up to carry out the assignment. On the other hand, we build up our understanding towards each other and strengthen our cooperation. It was also enhancing our knowledge on the topics involved in the assignment. First and foremost, we would like to wish our greatest thanks to God since he gave us the patience and calmness in doing the assignment well. As there is no building without basement, we had our lecturer, Puan Azyanzuhaila who had been guiding us to do the assignment as perfect as possible. She had given us some tips on how to make the assignment to be presentable and the details on how to carry on smoothly with the assignment. Last but not least, we would like to show our gratitude to our parents who support us economically. We would like to thank our friends who really shares information with us without showing any depression. We had put our full effort to complete the assignment as we wished. May the work gives the best satisfaction. “Best cooperation is the key to success” Thank You.
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ACKNOWLEDGEMENT
We would like take this wonderful opportunity to wish a lot of thanks from our bottom of heart. We have learnt many things directly and indirectly throughout this mathematics coursework. All the more, we gained many extra knowledge and it was a great fun of doing such a coursework as it involves something regarding our daily life. We practised to be punctual when a meeting was put up to carry out the assignment. On the other hand, we build up our understanding towards each other and strengthen our cooperation. It was also enhancing our knowledge on the topics involved in the assignment.
First and foremost, we would like to wish our greatest thanks to God since he gave us the patience and calmness in doing the assignment well. As there is no building without basement, we had our lecturer, Puan Azyanzuhaila who had been guiding us to do the assignment as perfect as possible. She had given us some tips on how to make the assignment to be presentable and the details on how to carry on smoothly with the assignment.
Last but not least, we would like to show our gratitude to our parents who support us economically. We would like to thank our friends who really shares information with us without showing any depression.
We had put our full effort to complete the assignment as we wished. May the work gives the best satisfaction.
“Best cooperation is the key to success”
Thank You.
ARITHMETIC SEQUENCE
1. Mr.Khalid buys a Home Teather System for his new house. Every month he pays his monthly payment. Determine how long it will take to repay a debt of RM7975 by monthly payments of RM10 initially increasing by RM5 each month. How much is the final payment?
Solution:
The monthly payments in RM are10, 15, 20,..........................
which is an arithmetic progression.
The formula for the sum of an A.P. involving the first n terms is:
Sn = n [2a + (n – 1)d] 2
The first payment is RM10, a = 10. The increment is RM5, d = 5. The sum of the payments is the debt of RM7975, Sn = 7975.Substituting into the formula,
7975 = n [2(10) + (n – 1)(5)]2
15950 = n(5n + 15)
n2 + 3n – 3190 = 0
(n – 55)(n + 58) = 0
n = 55 [n = -58 is rejected]
Therefore, the time taken to repay the debt is 55 months.
The final payment or the 55th payment
U55 = a + (n – 1)d
= 10 + (55 – 1)(5) = 280
Therefore, the final payment is RM280.
Solution by graphic calculator for A.P To use the TI-84 PLUS for the Arithmetic sequences and series, we need to change the mode to sequential (Seq)
Press MODE , select Seq
Press Y= and we can notice that this display is different than the display for Y= from the mode FUN
From this display we find
a) Nmin = the smallest termb) u(n) = formula for nth term
= (n-1)th term + difference= u(n-1) + difference
c) un(min) = the first term
Now, key in the formula for Tn
Key in u(n) = as 2nd [u] ( X, T, θ, n - 1 ) + 5
Key in u(nMin) = 10
To find the sum of S26 from the arithmetic series above, we can create a summation formula in v(n)
Sum of the nth term = sum to the term (n-1) + nth term
v(n) = v(n – 1) + u(n – 1) + 5
and v(nMin) = 10
Key in the settings in TI-84 PLUS
Press 2nd [table] and move the cursor to find the value S55
From this table we can find the time taken to repay the debt is 55 months when the S55 shows v(n)=7975
We also can find the final payment of T55 RM 280.00 u(n)
GEOMETRIC SEQUENCES
2. Mr. Razak has planned to invest in an insurance company. If RM100 is invested each year at 5% interest compounded annually, what would be the total amount of the investment Mr. Razak has made after 10 years (before the 11th deposit is made)?
After 1 year the amount invested will have added to it the interest for the year. Therefore, for the last (10th) RM100 invested, its value will become
RM100(1 + 0.05) = RM100(0.05) = RM105
The next to last RM100 will have interest added twice. After 1 year its value becomes RM100(1.05), and after 2 years it is RM100(1.05)(1.05) = RM100(1.05)2. In the same way, the value of the first RM100 becomes RM100(1.05)10, since it will have interest added 10 times. This means that we are to find the sum of the sequences