sorry for before it is my first time using this website and this is the real 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?

t1 = 23 ---> a = 23

t3 = 92 --->ar^2 = 92
divide them
r^2 = 4
r = ± 2

sum(n) = a(r^n - 1)/(r-1) = 62813

If r = 2
23( 2^n - 1)/(2-1) = 62813
2^n - 1 = 2731
2^n = 2732

but ....
n has to be a whole number, since it stands for the number of terms
2732 is NOT a power of 2,
so even though you typed it again with a correction , the question is still flawed.

Answer is 13 terms

No problem! I can help you with your question. To find the number of terms in the geometric series, we first need to determine the common ratio (r) and then use it to calculate the number of terms.

In a geometric series, each term is obtained by multiplying the previous term by a constant ratio, denoted as 'r'.

We are given:
t1 = 23 (first term)
t3 = 92 (third term)
Sum of all terms = 62813

To find the common ratio (r), we can use the formula for the nth term of a geometric series:
tn = t1 * r^(n-1)

Let's start by finding 'r' using the given information:
t3 = t1 * r^(3-1)
92 = 23 * r^2

Divide both sides of the equation by 23:
4 = r^2

Taking the square root of both sides, we get:
r = 2 or r = -2

Since geometric series can't have negative common ratios (as it would alternate positive and negative terms), we choose r = 2.

Now that we have the common ratio (r = 2), we can find the number of terms using the formula for the sum of a geometric series:

sum = t1 * (1 - r^n) / (1 - r)
62813 = 23 * (1 - 2^n) / (1 - 2)

To simplify, we have:
62813 = 23 * (1 - 2^n) / (-1)

Multiply both sides by -1 to eliminate the negative sign:
-62813 = 23 * (1 - 2^n)

Divide both sides by 23 to isolate the term (1 - 2^n):
-2727 = 1 - 2^n

Rearrange the equation:
2^n = 1 + 2727

Subtract 1 from both sides:
2^n = 2728

Now, let's solve for n. Take the logarithm (base 2) of both sides:
n = log2(2728)

Using a calculator, we find:
n ≈ 11.42

Since the number of terms must be a whole number, we round up to the nearest integer:
n = 12

Therefore, there are 12 terms in the geometric series.