Question

in a geometric series t1=23,t3=92 and the sum of all of the terms of the series is 62813. How many terms are in the series

Answer 1

$t_1=23,\ \ \ t_3=92\\\\the\ geometric\ series\ \ \ \Rightarrow\ \ \ t_3=t_1\cdot q^2;\ \ \ q- the\ quotient \\\\92=23\cdot q^2\ /:23 \ \ \Rightarrow\ \ \ q^2= 4\ \ \ \Leftrightarrow\ \ \ (q=2\ \ \ or\ \ \ q=-2)\\\\Sum\\S_n=t_1\cdot \frac{\big{1-q^n}}{\big{1-q}} \ \ \ \Rightarrow\ \ \ 62813=23\cdot \frac{\big{1-q^n}}{\big{1-q}} \ /:23 \ \ \ \Rightarrow\ \ \2731= \frac{\big{1-q^n}}{\big{1-q}}$

$1)\ \ \ q=2\\\Rightarrow\ \ \ 2731= \frac{\big{1-2^n}}{\big{1-2}} \ \ \ \Rightarrow\ \ \ 2731=2^n-1\ \ \ \ \Rightarrow\ \ \ 2^n=2732\\\\\Rightarrow\ \ \ n=log_22732\ \notin\ Natural$

$2)\ \ q=-2\ \ \ \ \Rightarrow\ \ \ 2731= \frac{\big{1-(-2)^n}}{\big{1-(-2)}} \ \ \ \Rightarrow\ \ 2731=\frac{\big{1-(-2)^n}}{\big{3}}\ /\cdot3 \\\\\ \ \ \Rightarrow\ \ \ 8193=1-(-2)^n\ \ \ \Rightarrow\ \ \ (-2)^n=-8192\ \ \ (\Leftrightarrow\ \ \ n-odd\ number)\\\\\ \Rightarrow \ \ \ (-2)^n=(-2)^{13}\ \ \ \Leftrightarrow\ \ \ n=13\\\\Ans.\ In\ the\ geometric\ series\ are\ 13\ terms.$