Math core term 3 project sequences, limits and differentiations Done by: -Fatima Salem -Safeya Mohammed Section: 12.52

What are sequences? A sequence is a list of numbers. Any time you write numbers in a list format, you are creating a sequence. Something as simple as 1, 2, 3, 4, 5, 6,.. is a sequence. Rather than just listing the numbers, we usually identify it as a sequence with the notation a n = 1, 2, 3, 4, 5, 6,... Usually there is some type of pattern to a sequence. In the sequence above, you are adding one to each term to get the next term. Terms can be identified by their location. We note the 1 st term in a sequence as a 1 and we would call the 5 th term in the sequence a 5. We described the pattern in the sequence as adding one to each term to get the next term. We can express this as a recursive formula by writing a n = a n-1 + 1

Types of sequences: There are two types of sequences: - Arithmetic Sequences - Geometric Sequences An arithmetic sequence : any time you are adding the same number to each term to complete the sequence, it is called an arithmetic sequence A geometric sequence: When you multiply every term by the same number to get the next term in the sequence, you have a geometric sequence

Task 1 Ahmed,who is a grade 09 student, is planning to save money while he is studying at ATHS. Ahmed has 5,000 Dhs to start with, and he is planning to save 500 Dhs per month. a) What is the type of the sequence at which the money will grow? It is an arithmetic sequence. b) How much money will he save after 4 years? 12 months in a year => 12(4) = 48 months a n = a 1 +d(n-1) a 48 = (48-1) a 48 = AED

c)His brother has borrowed from him some money, his brother agreed to pay 50 Dhs in the first month and 25 Dhs more in each of the following months; how much did Ahmed’s brother borrow knowing that it took him 1 year to pay his debt? a n = a 1 +d(n-1) a 12 = (12-1) a 12 = 325 AED Ahmed’s brother borrowed 325 AED.

d) Let the amount saved at the end of the first year be and the amount that was saved at the second year, and the amount that was saved at the end of the fourth year, find a relation between ?

- The difference between the first year and the second year is 6000 AED - The difference between the second year and the Forth year is AED - Each year, the amount saved is increasing by 6000AED, a+6000=b b =c First year(a)=a 12 = a 1 +d(12-1) a 12 = (12-1) a 12 = AED Second year(b)=a 24 = a 1 +d(12-1) a 24 = (24-1) a 24 = AED Fourth year (c) = a 48= AED

Task 2 Ahmed wanted to invest his savings; he bought shares in a company, the company will offer an interest rate of 8% annually. a) How much money will be in the account in 4 years?

b) After 4 years the shares were decreased by 20% each year, after how many years Ahmed will lose all of the profits? 1 st year = (0.8) = nd year = (0.6) = rd year = (0.4) = th year = (0.2) = th year = (0.0) = Zero He will lose all of his profits in 5 years.

Task 3 1) Ahmed has started his own company. He plans to hire 8 new employees. He wants to hire males and females, knowing that there are an equal number of males and females applying for the job, use the binomial theorem to find the all combinations of males and females that could be hired.

Total :8, (m+f) 8 8C 0 m 8 = (1)m 8 = m 8 8C 1 m 7 f 1 = (8)m 7 f 1 =8m 7 f 8C 2 m 6 f 2 = (28)m 6 f 2 =28m 6 f 2 8C 3 m 5 f 3 = (56)m 5 f 3 =56m 5 f 3 8C 4 m 4 f 4 = (70)m 4 f 4 =70m 4 f 4 8C 5 m 3 f 5 = (56)m 3 f 5 =56m 3 f 5 8C 6 m 2 f 6 = (28)m 2 f 6 =28m 2 f 6 8C 7 m 1 f 7 = (8)m 1 f 7 =8mf 7 8C 8 f 8 = (1) f 8 =f 8

2) Ahmed established a phone chain in which every staff member calls two other staff member to notify them about company events, the first round of calls begins with Ahmed calling two members. If there are 94 total staff members, how many rounds of calls are there to pass the information to all employees? a n = a 1 r n-1 94= 1(2) n-1 94=2 n-1 we have to plug in till we reach 94. When n=7  the answer will be 64 When n=8  the answer will be 128 So, there will be 8 rounds.

3) Those employees will be seated around tables attached end-to-end for an event, How many tables are required to seat all the employees?

a1=4 a2=6 a3=8 d=2 Between 8 to 9 tables are required to seat all the 94 employees.

Derivatives The derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.

Applications of derivatives Rates of Change Critical Points Minimum and Maximum Values Finding Absolute Extrema

Task 4 The revenue of his company was modeled by ; where is the number of items sold. a) Write a Java program that gives the value of to fill the table below:

c) Let the cost to produce items is, find the profit when 20 items are sold.