Sequences & Series Formulas Flashcards

Arithmetic Sequence Formulas

• Nth Term Formula: ( a_n = a_1 + d(n-1) )

  • Represents the output for the nth term.
  • ( a_1 ) is the first term, ( d ) is the common difference, ( n ) is the term position.

• Summation Formula: ( S_n = \frac{n}{2}(a_1 + a_n) )

  • Calculates the sum of the first ( n ) terms of an arithmetic series.
  • ( S_n ) is the total sum, ( n ) is the number of terms, ( a_1 ) is the first term, and ( a_n ) is the last term.

Geometric Sequence Formulas

• Nth Term Formula: ( a_n = a_1 r^{n-1} )

  • Defines the output for the nth term.
  • ( a_1 ) is the first term, ( r ) is the common ratio, ( n ) indicates the term number.

• Summation Formula: ( S_n = a_1 \left(\frac{1 - r^n}{1 - r}\right) )

  • Used for finding the sum of the first ( n ) terms of a geometric series.
  • ( S_n ) denotes the total sum, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) signifies the number of terms.

Recursive Formulas

• Recursive Arithmetic Formula: ( a_n = a_{n-1} + d )

  • Expresses the nth term based on the previous term.
  • Adds the common difference ( d ) to the previous term ( a_{n-1} ).

• Recursive Geometric Formula: ( a_n = a_{n-1} \times r )

  • Indicates how to find the nth term using the preceding term.
  • Multiplies the previous term ( a_{n-1} ) by the common ratio ( r ).