Geometric Sequence Formulas and Term Calculations

Chapter 1: Term $a_n$ and Geometric Sequences

• Definition of sequence notation:

  • The variable $a_n$ represents the general or $n\text{th}$ term of a sequence.
  • The variable $a_1$ represents the first term of the sequence.

• Standard formula for a geometric sequence:

  • Equation: $a_n = a_1 \times r^{n-1}$
  • Components:
  • $a_n$: The $n\text{th}$ term of the geometric sequence.
  • $a_1$: The initial term.
  • $r$: The common ratio between consecutive terms.
  • $n-1$: The exponent applied to the ratio $r$.

Geometric Curve Formulations and Corrections

• Expression stated for geometric curves:

  • Stated equation: $a_n = a_1 + r \times (n-1)$
  • Components of the linear expression:
  • $a_n$: The resulting term value.
  • $a_1$: The initial term.
  • $r \times (n-1)$: The product of multiplier $r$ and position offset $(n-1)$.

• Noted error and formula modification:

  • The formula written as $a_n = a_1 + r \times (n-1)$ is an uncorrected form that requires changing.
  • A geometric relationship requires exponentiation $a_1 \times r^{n-1}$ rather than additive linear offset $a_1 + r \times (n-1)$.