Arithmetic and Geometric Sequences

Sequence

Arithmetic Sequences

• Definition: A sequence where the difference between consecutive terms is constant.
• Common difference (d): The fixed amount added to each term to get the next term.
• General formula:
  • ( a_n = a_1 + (n - 1) \cdot d )
  • Where ( a_n ) is the nth term, ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the term number.

Geometric Sequences

• Definition: A sequence where each term is obtained by multiplying the previous term by a constant.
• Common ratio (r): The fixed factor by which each term is multiplied.
• General formula:
  • ( a_n = a_1 \cdot r^{(n - 1)} )
  • Where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.

nth Term

• Refers to a general term in a sequence, allowing the calculation of any term based on its position (n).
• For arithmetic sequences: ( a_n = a_1 + (n - 1) \cdot d )
• For geometric sequences: ( a_n = a_1 \cdot r^{(n - 1)} )

Term-to-Term Rule

• A method for generating the next term in a sequence based on the current term.
• For arithmetic sequences: ( a_{n+1} = a_n + d )
• For geometric sequences: ( a_{n+1} = a_n \cdot r )

7th Term

• To find the 7th term in a sequence:
  • Arithmetic:
    • Use the formula: ( a_7 = a_1 + (7 - 1) \cdot d )
  • Geometric:
    • Use the formula: ( a_7 = a_1 \cdot r^{(7 - 1)} )
• Substitute the values of ( a_1 ), ( d ) (for arithmetic), or ( r ) (for geometric) into the formulas to calculate the 7th term.

Arithmetic Sequences

• An arithmetic sequence has a constant difference between consecutive terms.
• The common difference (d) represents the fixed value added to each term to produce the next term.
• The general formula for the nth term is ( a_n = a_1 + (n - 1) \cdot d ), where:
  • ( a_n ): nth term
  • ( a_1 ): first term
  • ( d ): common difference
  • ( n ): term number

Geometric Sequences

• A geometric sequence is formed by multiplying the previous term by a constant factor.
• The common ratio (r) is the fixed amount multiplied to generate each successive term.
• The general formula for the nth term is ( a_n = a_1 \cdot r^{(n - 1)} ), which includes:
  • ( a_n ): nth term
  • ( a_1 ): first term
  • ( r ): common ratio
  • ( n ): term number

nth Term Calculation

• The nth term represents a general term, enabling the calculation of any position in the sequence.
• For arithmetic sequences, ( a_n ) is calculated using ( a_n = a_1 + (n - 1) \cdot d ).
• For geometric sequences, ( a_n ) is derived from ( a_n = a_1 \cdot r^{(n - 1)} ).

Term-to-Term Rule

• The term-to-term rule defines how to generate the next term based on the current term.
• For arithmetic sequences, the next term is calculated as ( a_{n+1} = a_n + d ).
• For geometric sequences, the next term follows ( a_{n+1} = a_n \cdot r ).

Finding the 7th Term

• To determine the 7th term in a sequence, specific formulas need to be used.
• For arithmetic sequences, apply ( a_7 = a_1 + (7 - 1) \cdot d ).
• For geometric sequences, utilize ( a_7 = a_1 \cdot r^{(7 - 1)} ).
• Substitute appropriate values for ( a_1 ), ( d ) (arithmetic), or ( r ) (geometric) to find the 7th term.

Formulas For Nth Term

• Linear Sequences use the formula: nth term = a + (n-1)d
  • a is the first term of the sequence, while d represents the common difference between terms.
• Quadratic Sequences follow the formula: nth term = an² + bn + c
  • a, b, and c are constants derived from the initial terms of the sequence.

Patterns In Sequences

• Arithmetic Sequences have a constant difference; for example, the sequence 2, 4, 6 has a common difference of d = 2.
• Geometric Sequences maintain a constant ratio between terms; for instance, in the sequence 3, 9, 27, the ratio is r = 3.
• Fibonacci Sequence is characterized by each term being the sum of the two preceding terms, starting with 0, 1, 1, 2, 3, 5.

Nth Term

• The nth term indicates the value of a term within a sequence based on its position, denoted as n.
• Nth term formulas are essential for calculating future terms efficiently, eliminating the need to list all previous terms.

Term-Term Rule

• The Term-Term Rule articulates the connection between consecutive terms, typically expressed as: T(n) = T(n-1) + k
  • T(n) represents the nth term, while k signifies the difference or pattern between terms.

7th Term

• To calculate the 7th term, apply the appropriate nth term formula based on the sequence type:
  • For arithmetic sequences, use: T(7) = a + 6d
  • For geometric sequences, use: T(7) = ar^(6)
  • For quadratic sequences, utilize: T(7) = a(7)² + b(7) + c
• Simply substitute n = 7 into the respective formula to find the desired term.

Nth Term

• Defines a method to find the value of any term based on its position (n) in the sequence.
• Arithmetic Sequence Formula: nth term = a + (n - 1)d
  • a represents the first term.
  • d denotes the common difference.
• Geometric Sequence Formula: nth term = a * r^(n - 1)
  • a is the first term.
  • r indicates the common ratio.
• Applicable to various sequences, including arithmetic and geometric ones, and can extend to more complex sequences.

Term-Term Rule

• Establishes the relationship between consecutive terms in a sequence.
• Arithmetic Sequence Rule: Each term is calculated by adding a constant (d) to the last term.
  • Formulated as a(n) = a(n-1) + d.
• Geometric Sequence Rule: Each term results from multiplying the previous term by a constant (r).
  • Formulated as a(n) = a(n-1) * r.
• Aids in recognizing patterns within sequences, allowing predictions of future terms.

7th Term Calculation

• To find the 7th term, use the specific sequence's nth term formula.
• Substitute n = 7 into the established formula for the sequence.
• Example for Arithmetic Sequence:
  • Given a = 3 and d = 2:
    • Calculation: 7th term = 3 + (7 - 1) * 2 = 15.
• Example for Geometric Sequence:
  • Given a = 2 and r = 3:
    • Calculation: 7th term = 2 * 3^(7 - 1) = 1458.

Summary

• Grasping nth terms and term-term rules is crucial for evaluating sequences effectively.
• The process to compute specific terms, like the 7th term, relies on these foundational principles.

This quiz covers the fundamentals of arithmetic and geometric sequences. You'll explore definitions, common differences, and ratios, as well as the general formulas for finding the nth term in each type of sequence. Test your understanding and enhance your skills in sequencing!