Answered

Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

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

Sagot :

kate200468
[tex]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}} \\\\[/tex]

[tex]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\\\\[/tex]

[tex]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.[/tex]

Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.