In a geometric series, [tex]t_1 = 23[/tex] and [tex]t_3 = 92[/tex], and the sum of all the terms of the series is 62,813. How many terms are in the series?

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.

Answered by kate200468 | 2024-06-10

To find the number of terms in a geometric series, we need to find the common ratio (r) and the first term (a). Using the given information and formulas for geometric series, we can find the number of terms in the series. There are approximately 922 terms in the series. ;

Answered by Burger1123 | 2024-06-18

In the geometric series with the first term 23 and the third term 92, the number of terms is calculated to be 13. The common ratio was found to be -2, which is valid as the overall sum of the series fits. Thus, the series contains 13 terms in total.
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Answered by kate200468 | 2024-12-17