q1 math

• Sequence


  An arrangement of any objects or a set of numbers in a particular order followed by some rule
• Term


  Each number in a sequence
• a1


  Represents the first term of the sequence
• a2


  Represents the 2nd term of the sequence
• a3


  Represents the third term of the sequence
• an


  Represents the nth term of the sequence
• Finite sequence


  A sequence which has a last term
• Finite sequence


  • 10, 12, 14, 16, 18
• Infinite sequence


  A sequence which has no last term
• Infinite sequence


  • 5, 10, 15, 20, ...
• Kinds of sequences


  • Arithmetic sequence
  • Geometric Sequence
  • Harmonic Sequence
  • Fibonacci
• Arithmetic sequence


  A sequence where every term after the first is obtained by adding a constant called the common difference (d)
• Arithmetic sequence


  • 1, 4, 7, 10
  • 15, 11, 7, 3
• Common difference


  The constant added to each term to get the next term in an arithmetic sequence
• Finding the nth term of an arithmetic sequence


  an = a1 + (n - 1)d
• Arithmetic means


  The terms between any two nonconsecutive terms of an arithmetic sequence
• Inserting arithmetic means


  1. a1 = 5

  2. a2 = 9

  3. a3 = 13

  4. a4 = 17

  5. a5 = 21

  6. a6 = 25
• Geometric sequence


  A sequence where each term after the first is obtained by multiplying the preceding term by a nonzero constant called the common ratio
• Geometric sequence


  • 32, 16, 8, 4, 2
• Common ratio


  The constant multiplier between each term in a geometric sequence
• Geometric means


  Any term/terms between 2 nonconsecutive terms in a geometric sequence
• Inserting geometric means


  1. a1 = 5

  2. a2 = 25

  3. a3 = 125

  4. a4 = 625

  5. a5 = 3125
• Finite geometric series


  The sum of the first n terms of a geometric sequence
• Finding the sum of a finite geometric series

  1. Sn = a1(1 - rn) / (1 - r), r ≠ 1

  2. Sn = na1, if r = 1
• Infinite geometric series


  The sum of an infinite number of terms in a geometric sequence
• Finding the sum of an infinite geometric series

  1. S∞ = a1 / (1 - r), when |r| < 1 (convergent)

  2. The sum cannot be determined when |r| ≥ 1 (divergent)
• Harmonic sequence


  A sequence such that the reciprocal of the terms form an arithmetic sequence
• Fibonacci sequence


  A sequence invented by Italian Leonardo Pisano Bigollo (1180-1250), where each term is the sum of the two preceding terms
• Fibonacci sequence


  • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
• Convergent geometric series


  When |r| < 1
• Sum to infinity of a convergent geometric series


  S∞ = a1/(1-r)
• Convergent geometric series


  • 1/3 + 1/9 + 1/27 + 1/81 + ... (r = 1/3)
  • 4/7 + 2/7 + 1/7 + 1/14 + ... (r = 1/2)
• Divergent geometric series


  When |r| ≥ 1, the sum cannot be determined because it will tend to infinity
• Divergent geometric series


  • 8 + 24 + 72 + 216 + ... (r = 3)
  • 3.5 + 10.5 + 31.5 + 94.5 + 283.5 + ... (r = 3.5)
• Fibonacci sequence


  A sequence invented by Italian Leonardo Pisano Bigollo (1180-1250), where each term is the sum of the two preceding ones
• The Fibonacci sequence was the outcome of a mathematical problem about rabbit breeding that was posed in the Liber Abaci
• Word problem: Mila's altitude after 10 hours

  1. a_n = a_1 + (n-1)d

  2. a_1 = 40, d = 10, n = 10
• Word problem: Man's salary after 5 years

  Salary = 60,000(1 + 0.05)^5 = 76,577
• Polynomial expression


  An expression of the form a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0
• Polynomial division (long division method)


  Dividing one polynomial by another