Arithmetic Sequences and Series

Sums and Terms

How to Find Terms

• To find a term in a sequence, use the formula: an = a1 + (n-1)d
• Where an is the term you're looking for, a1 is the first term, n is the term number, and d is the common difference

Common Difference

• The common difference is the difference between each term in a sequence
• It is denoted by d and can be found using the formula: d = an - an-1
• A sequence with a common difference is called an arithmetic sequence

Sequence

• A sequence is a list of numbers in a specific order
• Each number in the sequence is called a term
• Sequences can be finite or infinite
• Types of sequences:
  • Arithmetic sequence: has a common difference between terms
  • Geometric sequence: has a common ratio between terms

Algebraic Expression

• An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations
• It can be used to represent a sequence or a series
• Examples of algebraic expressions:
  • 2n + 1
  • 3n^2 - 2
  • n^2 + 2n - 3
• Algebraic expressions can be used to find the sum of a sequence or series

Sums and Terms

Finding Terms

• Use the formula an = a1 + (n-1)d to find a term in a sequence
• Where an is the term you're looking for, a1 is the first term, n is the term number, and d is the common difference

Common Difference

• The common difference is the difference between each term in a sequence
• It is denoted by d and can be found using the formula: d = an - an-1
• An arithmetic sequence has a common difference between terms

Sequences

• A sequence is a list of numbers in a specific order
• Each number in the sequence is called a term
• Sequences can be finite or infinite
• Types of sequences:
  • Arithmetic sequence: has a common difference between terms
  • Geometric sequence: has a common ratio between terms

Algebraic Expressions

• An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations
• It can be used to represent a sequence or a series
• Examples of algebraic expressions:
  • 2n + 1
  • 3n^2 - 2
  • n^2 + 2n - 3
• Algebraic expressions can be used to find the sum of a sequence or series

Sums and Terms

• To find a term in a sequence, use the formula: an = a1 + (n-1)d
• The formula consists of:
  • an: nth term
  • a1: first term
  • n: term number
  • d: common difference
• Example: Find the 10th term of a sequence with a first term of 3 and a common difference of 2: an = 3 + (10-1)2 = 3 + 18 = 21

Common Difference

• The common difference (d) is the constant difference between consecutive terms in an arithmetic sequence.
• Formula to find the common difference: d = an - an-1
• Where:
  • an: nth term
  • an-1: (n-1)th term
• Example: Find the common difference of a sequence with terms 2, 5, 8, 11,... : d = 5 - 2 = 3

Sequences

• A sequence is an ordered list of numbers.
• Types of sequences:
  • Arithmetic sequence: each term is obtained by adding a fixed constant to the previous term.
  • Geometric sequence: each term is obtained by multiplying the previous term by a fixed constant.

Algebraic Expressions

• An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations.
• Examples of algebraic expressions:
  • 2x + 3
  • x^2 - 4
• To evaluate an algebraic expression, substitute the value of the variable into the expression.
• Example: Evaluate 2x + 3 when x = 4: 2(4) + 3 = 11