AI Made - Part 1 - Week 12 to 13

• What is a sequence in mathematics?

  A discrete structure representing an ordered list
• What is each number in a sequence called?

  Term
• How is each term in a sequence denoted?

  an where n is the index
• What does the notation a1, a2, a3, ..., an represent?

  Terms in a sequence
• What are applications of sequences?

  • Algorithm Analysis
  • Data Structures
  • Networking
  • Database System
  • Computer Graphics
  • Machine Learning
• What is an explicit formula in a sequence?

  A formula defining the n-th term of a sequence
• What is the explicit formula for the sequence generated by $a_n =$$2n$?


  $2n$
• What is the value of $a_5$ in the sequence defined by $a_n =$$2n$?


  $10$
• What are the two parts of a recursive formula?

  Initial condition and recurrence relation
• What is the initial condition for the sequence 2, 5, 8, 11, ...?

  $a_1 =$$2$
• How do you find the 3rd term in the sequence defined by $a_n =$$a_{n-1} +$$3$?


  Calculate $a_2$ first
• What are the first five terms of the sequence defined by $Y(n) =$$2n^2 +$$3$?


  5, 11, 21, 35, 53
• What are the two major types of sequences?

  • Arithmetic
  • Geometric
• What defines an arithmetic sequence?

  Each term is calculated by adding a constant
• What is the common difference in an arithmetic sequence?

  The constant between consecutive terms
• How do you calculate the common difference (d) in an arithmetic sequence?

  $d =$$a_n - a_{n-1}$
• What is the formula for the n-th term of an arithmetic sequence?

  $a_n =$$a_1 +$$(n-1)d$
• What are the first five terms of the arithmetic sequence defined by $a_n =$$1 +$$(n-1)4$?


  1, 5, 9, 13, 17
• What is the 32nd term of the arithmetic sequence starting with 79 and having a common difference of -4?

  $-45$
• What is the formula for the sum of the first n terms in an arithmetic sequence?

  $S_n =$$\frac{n(a_1 + a_n)}{2}$
• What is the sum of the first five terms of the arithmetic sequence 1, 5, 9, 13, 17?

  $45$
• What defines a geometric sequence?

  Each term is calculated by multiplying a constant
• What is the common ratio in a geometric sequence?

  The constant between consecutive terms
• What is the formula for the n-th term of a geometric sequence?

  $a_n =$$a_1 \cdot r^{n-1}$
• What are the first five terms of the geometric sequence with $a_1 =$$2$ and $r =$$3$?


  2, 6, 18, 54, 162
• What is the 10th term of the geometric sequence starting with 2 and having a common ratio of 3?

  $39,366$
• What is the formula for the sum of the first n terms in a geometric sequence?

  $S_n =$$\frac{a_1(1 - r^n)}{1 - r}$
• What is the sum of the first five terms of the geometric sequence 2, 6, 18, 54, 162?

  $242$
• How do you find the term number (n) for a specific term in the geometric sequence 1, 2, 4, 8, 16, 32, ...?

  Use logarithms to solve
• What is the sum of the geometric series for the sequence 1, 2, 4, 8, 16, 32, ..., up to 15 terms?

  $32,767$
• What are the differences between arithmetic and geometric sequences?

  • Arithmetic: Add a constant (common difference)
  • Geometric: Multiply by a constant (common ratio)