If A=1+r^a+r^2a+r^3a+.......................to infinity
B=1+r^b+r^2b+................to infinity
then find a/b=?????
Is r^2a supposed to be r^(2a) or a*r^2? In other words, is the a part of the exponent?
what the heck is that?
exponents are like 2 cubed or 2 squared.
2 cubed= 8
2 squared= 4
exponents are like 2 cubed or 2 squared.
2 cubed= 8
2 squared= 4
because cubed means three, and squared means two. so 2 squared would be 2*2=4 and two cubed would be 2*2=4*2=8.
In the given expression A = 1+r^a+r^(2a)+r^(3a)+..., it appears that the exponent a is part of the term. Each subsequent term is obtained by multiplying the previous term by r^a.
Similarly, in the expression B = 1+r^b+r^(2b)+..., the exponent b seems to be part of the term. Each subsequent term is obtained by multiplying the previous term by r^b.
To find the ratio a/b, we can divide the two expressions:
A / B = (1+r^a+r^(2a)+r^(3a)+...) / (1+r^b+r^(2b)+...)
Now, since both A and B are infinite geometric series, we can use the formula for the sum of an infinite geometric series to simplify this expression. The formula is given as:
Sum = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
Let's assume that the first term of both series A and B is 1 (which is given in the expressions), and the common ratios are r^a and r^b respectively.
Thus, applying the formula to both A and B, we get:
A / (1 - r^a) and B / (1 - r^b).
Now, to find the ratio a/b, we divide A / (1 - r^a) by B / (1 - r^b):
(a/b) = (A / (1 - r^a)) / (B / (1 - r^b))
By simple algebraic manipulation, we can rewrite it as:
a/b = (A * (1 - r^b)) / (B * (1 - r^a))
So, the ratio a/b is equal to (A * (1 - r^b)) / (B * (1 - r^a)).