Series - Geometric series

Initial term : \(a_1\)

\(N^{th}\) term : \(a_n\)

Common ratio : q

Number of terms in the series : n

Sum of thefirst n terms : \(S_n\)

Sum to infinity : S

 

  1. \(a_n= qa_{n-1}= a_1q^{n-1}\)
  2. \(a_1.a_n=a_2.a_{n-1}=...=a_i.a_{n+1-i}\)
  3. \(a_i= \sqrt {a_{i-1}.a_{i+1}}\)
  4. \(S_n= {a_nq-a_1 \over q-1}= {a_1(q^n-1) \over q-1 }\)
  5. \(S= \displaystyle\lim_{n \to \infty}S_n = {a_1 \over 1-q }\)
    For \(|q|<1,\)the sum S converges as \(n \to \infty .\)