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

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.

Step-by-step explanation:

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, we can solve for these values.

1. First, we need to find the common ratio (r). We can do this by dividing the third term (t3) by the first term (t1): r = t3/t1 = 92/23 = 4.
2. Next, we can find the first term (a) by substituting the values we know into the formula for the nth term of a geometric series: t1 = a * r^(1-1). Since r = 4 and t1 = 23, we can solve for a: 23 = a * 4^(1-1) => 23 = a * 1 => a = 23.
3. Now that we have the first term (a) and the common ratio (r), we can use the formula for the sum of a geometric series: S = a * (1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. Substituting the values we know, we can solve for n: 62813 = 23 * (1 - 4^n) / (1 - 4) => 62813 = 115 * (1 - 4^n) => 4^n - 1 = 115/23 => 4^n - 1 = 5 => 4^n = 6 => n = log4(6) = 921.996 (rounded to the nearest whole number).

Therefore, there are approximately 922 terms in the series.

Answer 2

`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\ \\otin\ 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.`