ISU: Sequences and Series

ISU: Sequences and Series HusamAbdulnabi, Yash Dave, Yahya Adam, Neena Raj

Sequences Arithmetic ,Geometric, and Recursive

Arithmetic Sequence • Each successive term is formed by adding the same number • Difference between consecutive terms is a constant • This constant is referred to as the common difference (d)

Arithmetic Sequence General Term • Arithmetic sequence is modeled after the following formula: In which: • tnis the term number • a is the first term • n is the number of terms in the sequence • d is the common difference

Examples a) Given 1,2,4,7… is it arithmetic? b) Given 6,11,16,21,… Find c) Given…7,10,13,16…and determine the first term (a) d) Given 2,9,16,…191, how many terms are in the sequence?

Rafiki’s Beads • Rafiki secretly collects beads for his staff. He usually keeps track, but Zazu stole his records that keeps track of his totals after every day (n): 1) Find his initial amount of beads (a) and the constant difference (d) 2) Determine and see how much beads he would have after 1 year from his initial amount.

Geometric Sequence • Each successive term is formed by multiplying by the same number. • Difference between consecutive terms is a common ratio (r) • The common ratio is a constant • Obtained by dividing a term with the term before it (excluding the first term).

Geometric Sequence General Term • The formula of geometric sequence is modeled after: In which: • a is the first term • n is the number of terms in the sequence. • r is the common ratio

HYENA ATTACK ! • Simba’s trusty mole spies a lone hyena lurking in the Pride Lands. Alarmed, the mole decides to observe the hyena’s actions. Suddenly there are 4 hyenas, then 16 and then 64. There’s a 5 minute interval between each hyena increase. Simba tells the mole it will take 25 minutes to gather the pride. • If the hyena numbers continue to increase at the same rate, how many more hyenas will arrive when the pride gathers. • 1, 4, 16, 64,…

Recursive Sequence • Set of numbers in which there is a dependence of the proceeding terms on the preceding terms • Since there is a direct dependence of a term on the terms before it, one, two or more terms must be given • General term formula must be used to find any other terms • By using the other terms to substitute into this general term formula of course

Recursive Sequence Fibonacci Sequence • Modeled after:

Example • Timon wanted to find the first 4 terms of the sequence determined by the formula where • Pumba was convinced that there must be another way how the general term for this sequence can be represented! Help him state it in function notation, and sketch the graph of the equation.

Series Arithmetic and Geometric

Arithmetic Series • The sum of the terms in an arithmetic sequence General Term • The arithmetic series is modeled as the following formula(s):

Arithmetic Series In which: • Snis the sum number • tnis the term number • a is the first term • n is the number of terms in the sequence • d is the common difference

Examples a) Determine S15 for 11+14.5+18+21.5+… b) Find the sum of 13+25+37+…+217

Mischievous Zazu • Rafiki finally catches up with the bird, but before that pesky bird gives back the baboon’s work, Zazu intends to make 9 twirls in the air which increase by an average of 4 seconds each circle. If he takes 7 seconds for his second round, what is the total air time before he respectfully comes down?

Geometric Series • Is the sum of the terms in a geometric sequence General Term • Geometric Series represented by each of these formulae:

Geometric Series In which • a is the first term • n is the number of terms • r is the common ratio

Too Many Hyenas • After the mole told Simba how many more hyenas would gather after 25 minutes, Simba requested that the total number of hyenas be calculated. Zazu started to manually count all of the hyenas and was having an extremely hard time…Can you help Zazu count the hyenas?

Pascal’s Triangle • First row is called Row 0, first column is called Column 0. • t n , r • n = row number • r = column number

Pascal’s Triangle • The terms in the next row always begin and end with a 1 • Each successive term is the sum of the two terms above it. • Even rows have an odd number of terms and odd rows have an even number of terms • The sum of any row can be expressed as 2 to the exponent of the row number

Pascal’s Triangle Find the number of ways to spells Pride Rock P R R I I I D D D D E E E E E R R R R R R O O O O O C C C C K K K

Binomial Theorem • Displays how to calculate power of binomial without expanding • Instead, by using the Pascal’s triangle, the binomial theorem allows for quicker factoring • Ex: (a + b)4 • = a4+ 4a3b + 6a²b² + 4ab3+ b4

Binomial Theorem • Ex: (a + b)4 • = a4+ 4a3b + 6a²b² + 4ab3+ b4 • The coefficients of the expansion correspond to the row of Pascal’s triangle that is the same as the exponent on the expansion. (In this case, it is row 4).

Binomial Theorem • The first term of the polynomial is the first term in the binomial raised to the exponent of the binomial Ex: (a + b)4 • The second term of the polynomial is the first term of the binomial with one less exponent multiplied by the second term from the binomial with one more exponent

Binomial Theorem • Each term in the polynomial has the first term from the binomial losing 1 from the exponent after each term when going left to right • The second term in each polynomial has the second term from the binomial gaining 1 more exponent each time when going left to right.

Binomial Theorem • Ex: Nalais having a serious problem expanding the expression (3y+2)3. Its takes too long to factor and she feels that she keeps making mistakes. Show her how to factor using the binomial theorem so she can impress Simba.

ISU: Sequences and Series