Find r in a geometric sequence if A=2 and A10=1024
A = 2 , term(10= ar^9 = 1024
then 2 r^9 = 1024
r^9 = 512
take 9th root
r = 2
To find the common ratio, denoted as `r`, in a geometric sequence, we need to know the first term (`A`) and the tenth term (`A10`).
In this case, the first term is given as `A = 2` and the tenth term is `A10 = 1024`.
We can use the formula for the nth term of a geometric sequence:
An = A * r^(n-1)
where An represents the nth term, A is the first term, r is the common ratio, and n is the term number.
We can plug in the values for A and A10 to form two equations:
A10 = A * r^(10-1)
1024 = 2 * r^9
Now, we can simplify the equation and solve it for `r`. Divide both sides of the equation by 2:
512 = r^9
Next, take the ninth root of both sides to isolate `r`. Since the equation may have two real solutions, we take both the positive and negative roots:
r = ± (512)^(1/9)
Using a calculator, we can compute this expression:
r ≈ ± 1.7456444939
Therefore, the common ratio `r` in the geometric sequence is approximately ±1.7456 (rounded to 4 decimal places).