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Geometric Series

Formula

Finite Geometric Series

  • For rā‰ 1r \neq 1
  • a+ar+ar2+ar3+...+arnāˆ’1=āˆ‘k=0nāˆ’1ark=a(1āˆ’rn1āˆ’r)a + ar + ar^2 + ar^3 + ... + ar^{n - 1} = \displaystyle\sum_{k=0}^{n-1} ar^k = a(\frac{1 - r^n}{1 - r})
    • aa is the first term
    • rr is the common ratio
    • nn is the term you are looking for

Infinite Geometric Series

  • Absolute value for rr must be less than 11 for the series to converge
    • If r>1r \gt 1 then the sum is āˆž\infty
  • a+ar+ar2+ar3+...+arnāˆ’1=āˆ‘k=0āˆžark=a1āˆ’ra + ar + ar^2 + ar^3 + ... + ar^{n - 1} = \displaystyle\sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}

Math Series